Derivative Calculator — Free Step-by-Step Solutions for Any Function
Enter any function below — our Perfect Calculus Engine returns the exact derivative instantly with a full rule-by-rule step-by-step breakdown, interactive graph, critical points, and inflection point analysis. Used by 50,000+ students monthly. No account needed. Works on every device.
![Derivative Calculator — Step-by-step solutions showing d/dx[x²·sin(x)] = 2x·sin(x) + x²·cos(x) with interactive graph and critical point analysis](/images/derivative-calculator-og.jpg)
Enter any function — our tool detects and solves it instantly ↓
A derivative calculator computes the instantaneous rate of change — the derivative — of any mathematical function. Type your function into the box below, click Calculate, and get the exact answer with a full step-by-step explanation. It supports polynomial, trigonometric, exponential, logarithmic, implicit, and multivariable functions.
✅ Try these — just type them exactly as shown into the calculator:
📚 What Is a Derivative? Complete Beginner's Guide with Examples
What Is a Derivative in Calculus?
A derivative measures the instantaneous rate of change of a function at any given point. Think of your car's speedometer: it shows exactly how fast you are moving at that precise moment — not your average speed over a trip. In calculus, the derivative of f(x) tells you the exact slope of the function's graph at every value of x. Our derivative calculator computes these rates of change instantly for any function — polynomial, trigonometric, exponential, logarithmic, implicit, or multivariable — and shows every step of the working so you understand why the answer is correct, not just what it is.
The derivative is written as f'(x), dy/dx, or d/dx[f(x)]. It is formally defined as the limit of the difference quotient: f'(x) = limh→0 [f(x+h) − f(x)] / h. Our differentiation calculator uses a professional-grade symbolic Computer Algebra System (CAS) — not numerical approximation — to deliver mathematically exact results every time. Mastering derivatives is essential for calculus, physics, engineering, economics, machine learning, and data science.
📈 Real-World Examples
- Speed: How fast your position changes over time
- Growth Rate: How quickly investments or populations grow
- Slope: How steep a hill or graph is at any point
- Temperature Change: How quickly temperature rises or falls
🎯 Why Learn Derivatives?
- Essential for physics, engineering, and economics
- Used in machine learning and data science
- Helps optimize business decisions
- Fundamental to advanced mathematics
Understanding The Basic Concept
Imagine you're driving a car. Your position changes over time. The derivative of position with respect to time is your velocity (speed). If you're at position 0 miles at time 0 hours, and position 60 miles at time 1 hour, your average speed is 60 miles per hour.
✨ Key Insight
The derivative tells you the instantaneous rate of change - not just the average over time. It's like knowing your exact speed at 2:15 PM, not just your average speed for the whole trip.
Basic Derivative Rules Made Simple
Step-by-Step Learning Path
Start Simple
Begin with basic power rule and constants
Practice Regularly
Use our calculator to check your work
Visualize
Use graphs to understand slope changes
Apply
Solve real-world problems
Practice with These Examples
Click any example below to try it in our calculator. Watch the step-by-step solution to understand each rule in action.
💡 Pro Tip for Beginners
Don't just memorize rules - understand why they work. The power rule works because when you have xⁿ, each unit increase in x gets multiplied n times. The derivative measures how much extra you get from that tiny increase.
Use our calculator's step-by-step solutions and interactive graphs to visualize what's happening. See how the slope changes as you move along the curve. This visual understanding makes derivatives much easier to grasp.
Common Questions About Derivatives — Answered by Calculus Experts
Q: What is the difference between a derivative and a differential?
A derivative (f'(x) or dy/dx) is a ratio that gives the instantaneous rate of change — the slope of the tangent line. A differential (dy, dx) represents an infinitesimally small change in the variable. The two are related: dy = f'(x) · dx. When you use this calculator, you are finding the derivative dy/dx.
Q: How do I find the second derivative (d²y/dx²)?
Select 2nd from the "Derivative Order" dropdown above, enter your function, and click Calculate. The second derivative measures the rate of change of the first derivative — it tells you the concavity of the function and is used to identify local maxima and minima via the Second Derivative Test.
Q: Why is the derivative of a constant zero?
Because constants don't change! A constant like 5 has a rate of change of 0 — it never gets larger or smaller. This is the Constant Rule: d/dx[c] = 0 for any constant c. Our step-by-step solutions clearly show when this rule is applied.
Q: What is implicit differentiation and how do I use it?
Implicit differentiation is used when y cannot be easily isolated — for example in x² + y² = 25. Enter this equation directly into our dy/dx calculator and it will automatically apply implicit differentiation, differentiating both sides with respect to x and solving for dy/dx.
Q: How do I differentiate a product of two functions?
Use the Product Rule: if f(x) = u(x) · v(x), then f'(x) = u'(x)·v(x) + u(x)·v'(x). Try the example "x³·sin(x)" above — our derivative solver will show each step of the product rule clearly. For functions like (x²+1)/(x-1), the Quotient Rule applies: f'(x) = [u'v − uv'] / v².
Q: Which differentiation rules does this calculator support?
Our differentiation calculator handles all standard rules: Power Rule, Constant Rule, Sum & Difference Rule, Product Rule, Quotient Rule, Chain Rule, Exponential & Logarithmic Rules, all trigonometric and inverse-trig derivatives, hyperbolic functions, logarithmic differentiation, and nth-order derivatives up to the 5th order.
Q: What is the derivative of sin(x)?
The derivative of sin(x) is cos(x). This is one of the fundamental trigonometric derivative results. To try it: type sin(x) into the calculator above and click Calculate — the result will be cos(x) with a full step-by-step explanation of the rule applied.
Q: What is the derivative of e^x?
The derivative of e^x is e^x — it is the only function that equals its own derivative. To verify: type e^x into the calculator and click Calculate. For a composite version, type e^(2*x) and the calculator will apply the chain rule automatically, returning 2·e^(2x) with all working shown.
Q: Can this calculator find second and third derivatives?
Yes. Simply select the derivative order — 2nd, 3rd, 4th, or 5th — from the Derivative Order dropdown before clicking Calculate. The calculator applies differentiation repeatedly and shows all intermediate steps. The second derivative f''(x) is particularly useful for the Second Derivative Test: if f''(c) < 0 at a critical point, it's a local maximum; if f''(c) > 0, it's a local minimum.
Q: What makes this a "Perfect Calculus Engine"?
Our engine uses a full symbolic Computer Algebra System (CAS) — built on math.js combined with the Nerdamer library — to compute exact symbolic derivatives, not rounded decimal approximations. Unlike basic calculators that only handle simple cases, our engine correctly processes complex nested expressions, implicit functions like x^2 + y^2 = 25, multivariable partial derivatives, and variable-power expressions like x^x. It automatically identifies the function type, applies every rule in the correct order, and simplifies the result to its cleanest symbolic form. This is the same approach used by professional mathematical software — delivered completely free, with no sign-up.
🧮 Derivative Calculator — Free Step-by-Step Differentiation Online
Enter any function into our free derivative calculator and instantly get the exact symbolic result with a complete rule-by-rule step-by-step solution. Whether you need to find the derivative of a polynomial, trigonometric, exponential, or logarithmic function — or tackle chain rule, product rule, quotient rule, implicit differentiation, partial derivatives, or higher-order derivatives — our Perfect Calculus Engine handles it all automatically. Built on a professional symbolic CAS (Computer Algebra System), it delivers mathematically exact results, not decimal approximations.
How to use: Type your function (e.g. x^2*sin(x)) → select order → click Calculate. For implicit differentiation enter x^2 + y^2 = 25. For integrals type integral(x^2, x). For limits use limit(sin(x)/x, x, 0). The engine auto-detects the type and applies the correct method.
x^x, nested functions, or when Fast Mode gives unexpected results.
- Click 🎯 Accurate Mode — wait for "✅ Accurate Mode ready" (loads once, ~5–15 sec on first use)
- Type or paste your function (e.g.
x^2 * sin(x)), then click Calculate or press Enter - After first load, subsequent calculations are instant — no more waiting
- Switch back to 🚀 Fast Mode anytime — completely silent, no popups
- Note: Physics formula solver always uses Fast Mode. All other operations (derivatives, integrals, limits, special functions) use the full SymPy engine in Accurate Mode.
💡 Smart Auto-Detection:
The calculator reads what you type and automatically picks the right method:
Derivative: sin(x), x^3*ln(x) → differentiates w.r.t. x |
Partial: x^2*y + y^3 → computes ∂/∂x and ∂/∂y |
Implicit: x^2 + y^2 = 25 → finds dy/dx
Integral: integral(x^2, x) or integral(sin(x), x, 0, pi) → indefinite or definite |
Limit: limit(sin(x)/x, x, 0) → evaluates (L'Hôpital if needed)
Special fn: gamma(5), erf(1), zeta(2) → special functions |
Physics: physics: F=m*a → physics formula solver
Built by DerivativeCalculus.com — Founded November 2025 by Mian Muhammad Asghar, Dubai UAE
Dual symbolic engine: Fast Mode (instant CAS) + Accurate Mode (Python-grade symbolic engine) · Verified against Stewart's Calculus & OpenStax · Last reviewed: March 2026
✅ 99.7% accuracy rate · 50,000+ monthly students · 150+ countries · 100% free — no sign-up ever
📚 Calculation History
📖 Advanced Derivative Calculator
Our advanced derivative calculator represents the next generation of mathematical computation tools. Unlike basic calculators, it provides comprehensive analysis including interactive 2D/3D graphing, higher-order derivatives (up to 5th order), numerical evaluation at specific points, and detailed function properties.
The calculator uses professional-grade algorithms to compute derivatives instantly while providing educational insights. It automatically detects function types (polynomial, trigonometric, exponential, logarithmic), identifies critical points, determines concavity, and provides comprehensive analysis that helps students understand not just the "what" but the "why" behind derivatives. If you are working on complex problems like chain rule or product rule, this tool simplifies the process.
Key advanced features include: Real-time LaTeX preview of your input, interactive graphing with tangent lines, 3D visualization for multivariable concepts, calculation history with localStorage persistence, and export capabilities for graphs and results. The interface is designed for both beginners and advanced users, with quick access tools and comprehensive options.
🎯 How to Use Advanced Features
Maximize your learning with these advanced features:
Advanced Feature Guide
- Higher-Order Derivatives: Use the order selector to calculate 2nd, 3rd, 4th, or 5th derivatives. Perfect for studying concavity and higher-order behavior.
- Interactive Graphs: Click and drag to explore functions. Use zoom controls to examine details. Toggle between function and derivative views.
- 3D Visualization: Explore multivariable concepts with our 3D graph feature. Rotate, zoom, and examine surfaces from different angles.
- Numerical Evaluation: Enter a specific x-value in the evaluation point field to get the exact derivative value at that point.
- Calculation History: Access your previous calculations via the history panel. Perfect for reviewing problems or continuing work.
- Export Tools: Save graphs as PNG images or export complete results as PDF reports for study or submission.
Pro Tips: Use the math toolbar for quick symbol insertion. Enable LaTeX preview for better equation visualization. Save complex calculations to history for quick reference. The calculator supports all standard mathematical functions including trigonometric (sin, cos, tan), exponential (e^x), logarithmic (ln, log), hyperbolic functions, and custom combinations.
🚀 How to Use This Derivative Calculator
Using our derivative calculator with steps is simple:
- Enter your function in the input field above
- Select the variable and derivative order
- Click "Calculate" to get instant results
- View detailed step-by-step solutions
- Explore interactive graphs and 3D visualizations
This free online derivative calculator supports all standard mathematical functions including trigonometric, exponential, logarithmic, and hyperbolic functions.
📏 Complete Guide to Differentiation Rules
Our derivative calculator supports every standard differentiation rule used in calculus — from the elementary power rule through logarithmic differentiation and implicit differentiation. Below is a comprehensive reference with formulas, plain-English explanations, and worked examples you can click to try instantly.
1. Power Rule — d/dx[xⁿ] = n·xⁿ⁻¹
The power rule is the most frequently used differentiation rule. When you differentiate a variable raised to a constant exponent, you multiply the coefficient by the exponent and reduce the exponent by one. It applies to all real-number exponents — positive, negative, and fractional.
2. Product Rule — d/dx[u·v] = u'v + uv'
The product rule applies whenever two differentiable functions are multiplied together. You differentiate the first factor (keeping the second unchanged), add the product of the original first factor with the derivative of the second. A common mnemonic: "derivative of first times second, plus first times derivative of second."
3. Quotient Rule — d/dx[u/v] = (u'v − uv') / v²
The quotient rule handles the derivative of one function divided by another. Numerator: derivative of top times bottom, minus top times derivative of bottom. Denominator: bottom squared. Mnemonic: "low dee-high minus high dee-low, over low-squared."
4. Chain Rule — d/dx[f(g(x))] = f'(g(x))·g'(x)
The chain rule is essential for differentiating composite functions — functions within functions. Differentiate the outer function (evaluated at the inner function), then multiply by the derivative of the inner function. This rule is applied automatically by our calculator for all nested expressions.
5. Logarithmic Differentiation
Logarithmic differentiation is a technique used when a function has a variable in both the base and the exponent — like xˣ — or when a product/quotient of many terms makes direct differentiation cumbersome. Take the natural log of both sides, differentiate implicitly, then solve for dy/dx.
6. Implicit Differentiation
Implicit differentiation is applied when y cannot be easily isolated on one side of the equation. Differentiate both sides of the equation with respect to x — treating y as a function of x — then collect all dy/dx terms on one side and solve. Our calculator detects implicit equations automatically when you enter something like x^2 + y^2 = 25.
🧮 Worked Derivative Examples — Step-by-Step Solutions
The following examples show common derivative problems solved in full. Click the Try in Calculator button on any example to load it and see the complete rule-by-rule step-by-step working generated by our engine.
Example 1: d/dx [x²]
Power RuleApply Power Rule: Bring the exponent (2) down as a coefficient, reduce the exponent by 1.
Result: f'(x) = 2x
Example 2: d/dx [sin(x)]
Trig DerivativeApply Standard Trig Rule: The derivative of sin(x) is the standard result cos(x).
Result: f'(x) = cos(x)
Example 3: d/dx [xˣ]
Logarithmic DifferentiationStep 1: Take natural log: ln(y) = x·ln(x)
Step 2: Differentiate both sides (implicit): (1/y)·y' = ln(x) + 1
Step 3: Multiply both sides by y: y' = x^x · (ln(x) + 1)
Result: d/dx[x^x] = x^x · (ln(x) + 1)
Example 4: d/dx [ln(sin(x))]
Chain Rule + Log RuleOuter function: ln(u) → derivative = 1/u
Inner function: u = sin(x) → u' = cos(x)
Apply Chain Rule: f'(x) = (1/sin(x)) · cos(x)
Result: f'(x) = cos(x)/sin(x) = cot(x)
Example 5: d/dx [e^(x²)]
Chain Rule + ExponentialOuter function: e^u → derivative = e^u
Inner function: u = x² → u' = 2x
Apply Chain Rule: f'(x) = e^(x²) · 2x
Result: f'(x) = 2x·e^(x²)
🌍 Real-World Applications of Derivatives
Derivatives are not just abstract mathematics — they power the technology, science, and economic systems that shape everyday life. Here is how differentiation is used across major fields.
Physics
Derivatives describe motion at its most fundamental level. Velocity is the derivative of position with respect to time (v = ds/dt). Acceleration is the derivative of velocity (a = dv/dt). Newton's second law, electromagnetic wave equations, and quantum mechanics all rely on differential calculus as their mathematical language.
Engineering
Engineers use derivatives to analyse stress and strain in structures, model heat transfer rates, optimise signal processing filters, and design control systems. The rate of change of electrical current (dI/dt) and voltage (dV/dt) are central to circuit analysis, while structural engineers use them to predict how beams deform under load.
Economics & Finance
In economics, marginal cost and marginal revenue are derivatives of the cost and revenue functions. Finding where MR = MC (profit maximisation) requires setting derivatives to zero. The Black-Scholes option pricing model uses partial derivatives called "the Greeks" (delta, gamma, theta, vega) to hedge financial risk.
Machine Learning & AI
Gradient descent — the algorithm that trains every neural network — is pure applied differentiation. At each training step, the model computes the partial derivative of the loss function with respect to every parameter (the gradient), then moves in the direction that reduces error. Without derivatives, modern AI would be impossible.
Optimisation
Finding the maximum or minimum of any function requires setting its derivative equal to zero (f'(x) = 0) to locate critical points, then using the second derivative test (f''(x) > 0 for a minimum, f''(x) < 0 for a maximum). This technique is used to minimise material costs, maximise production output, and optimise routing algorithms.
Biology & Medicine
Pharmacokinetics uses derivatives to model how quickly drugs are absorbed and eliminated from the body. Population growth models rely on differential equations — which are built on derivatives. In epidemiology, the rate of change of infection spread (dI/dt) drives predictions used in public health planning.
Whenever a quantity changes — and you need to know the rate of that change — derivatives are the tool. Use our derivative calculator to explore the mathematics behind any real-world model by entering its function and computing d/dx instantly.
📐 Advanced Derivative Formulas
Our advanced calculator handles these complex differentiation formulas automatically:
Higher-Order Rules
- 2nd Derivative: f''(x) = d²/dx²[f(x)]
- Leibniz Notation: dⁿ/dxⁿ[f(x)·g(x)]
- General Power: dⁿ/dxⁿ[xᵐ] = m!/(m-n)!·xᵐ⁻ⁿ
- Trig Higher: dⁿ/dxⁿ[sin(x)] = sin(x + nπ/2)
Advanced Rules
- Implicit: d/dx[f(y)] = f'(y)·dy/dx
- Parametric: dy/dx = (dy/dt)/(dx/dt)
- Inverse: d/dx[f⁻¹(x)] = 1/f'(f⁻¹(x))
- Logarithmic Diff: d/dx[y] = y·d/dx[ln(y)]
Special Functions
- Hyperbolic: d/dx[sinh(x)] = cosh(x)
- Inverse Trig: d/dx[arcsin(x)] = 1/√(1-x²)
- Gamma: d/dx[Γ(x)] = Γ(x)·ψ(x)
- Bessel: d/dx[Jₙ(x)] = ½[Jₙ₋₁(x)-Jₙ₊₁(x)]
New to Derivatives?
Start with our Interactive Derivative Rules Hub to learn the fundamentals with 50+ examples and practice problems.
⭐ Why Choose Our Advanced Calculator?
Comparison of advanced features across popular derivative calculators:
❓ Frequently Asked Questions — Derivative Calculator
Everything you need to know about using our free derivative calculator, how derivatives work, and how to apply differentiation rules — answered by calculus experts.
What is a derivative calculator?
A derivative calculator is a free online tool that computes the exact derivative — the instantaneous rate of change — of any function. Type a function such as x²·sin(x) or e^x, click Calculate, and instantly get the exact symbolic result with a complete step-by-step rule explanation, interactive graph, and critical point analysis. No sign-up required.
Can the calculator solve derivatives step by step?
Yes — completely. Every calculation includes a full step-by-step solution showing exactly which rule was applied at each stage (power rule, chain rule, product rule, quotient rule, etc.) and the mathematical justification. Each step is numbered and colour-coded by rule type, so you can follow the working from start to finish. This makes our calculator ideal for students checking homework, teachers creating examples, and anyone who wants to understand why the answer is correct.
Does it support trigonometric functions?
Yes, fully. Our differentiation calculator supports all six standard trig functions (sin, cos, tan, cot, sec, csc), all six inverse trig functions (arcsin, arccos, arctan, arccot, arcsec, arccsc), and all six hyperbolic functions (sinh, cosh, tanh, coth, sech, csch). Composite trig expressions like sin(x²), ln(cos(x)), and tan(e^x) are handled automatically via the chain rule.
Can it calculate implicit derivatives?
Yes. Enter an equation such as x^2 + y^2 = 25 or x^3 + y^3 = 6*x*y directly into the input field. The calculator automatically detects the implicit form, differentiates both sides with respect to x (treating y as a function of x), collects all dy/dx terms, and solves for dy/dx — showing every step of the implicit differentiation process.
What is the derivative of sin(x)?
The derivative of sin(x) is cos(x). Written formally: d/dx[sin(x)] = cos(x). This is a fundamental trigonometric derivative result. Type sin(x) into the calculator above and click Calculate to see the full step-by-step working.
What is the derivative of e^x?
The derivative of e^x is e^x — the only function equal to its own derivative. For composites, the chain rule gives d/dx[e^(f(x))] = e^(f(x))·f'(x). Example: d/dx[e^(2x)] = 2e^(2x). Type e^(2*x) into the calculator to see the full chain rule working.
Can I calculate second, third, and higher-order derivatives?
Yes. Select the derivative order — 2nd, 3rd, 4th, or 5th — from the Derivative Order dropdown before clicking Calculate. The engine applies differentiation repeatedly and shows every intermediate step for each order. The second derivative f''(x) is particularly useful for the Second Derivative Test: f''(c) < 0 indicates a local maximum; f''(c) > 0 indicates a local minimum; f''(c) = 0 requires further analysis. The third derivative (jerk) is used in physics and motion analysis.
What differentiation rules does this calculator support?
Our derivative calculator supports all standard differentiation rules: Power Rule, Constant Rule, Sum and Difference Rule, Constant Multiple Rule, Product Rule, Quotient Rule, Chain Rule, Exponential Rule (e^x and a^x), Natural Logarithm Rule (d/dx[ln(x)] = 1/x), all trigonometric derivatives, all inverse trig derivatives, hyperbolic function derivatives, Logarithmic Differentiation (for functions like x^x), Implicit Differentiation, and Partial Differentiation for multivariable functions.
How accurate is this derivative calculator?
This derivative solver uses a dual symbolic Computer Algebra System (CAS) — Fast Mode for instant results and Accurate Mode (powered by a full Python-grade symbolic engine) for maximum precision. Both modes deliver exact symbolic derivatives, not rounded decimal approximations. Results like 2x·cos(x²) rather than imprecise floating-point numbers. This is the same class of symbolic engine used in professional mathematical software, making it one of the most reliable online derivative calculators available.
Is this derivative calculator completely free?
Yes — completely and permanently free with no sign-up, no subscription, and no limits on calculations. Every feature — step-by-step solutions, interactive graphs, higher-order derivatives (up to 5th), implicit differentiation, partial derivatives, integrals, and limits — is available at zero cost. Unlike Symbolab, which locks step-by-step solutions behind a paywall, or Wolfram Alpha Pro, which charges for full working, everything on DerivativeCalculus.com is free forever.
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Master calculus with our comprehensive collection of free learning resources. From beginner guides to advanced techniques, we provide everything you need to excel in derivatives and calculus.
Comprehensive Guides
- ▶ What is a Derivative?
- ▶ Derivative Rules Masterclass
- ▶ Real-World Applications
- ▶ Implicit Differentiation Guide
Practice Materials
Start Practicing✨ Featured Resource
Derivative Rules Cheat Sheet
Download our comprehensive cheat sheet covering all derivative rules including power rule, product rule, quotient rule, chain rule, trigonometric, exponential, and logarithmic derivatives.
All resources are 100% free forever. No sign-up required. No limits on access.
🚀 Explore All ResourcesCreate unlimited free math worksheets in seconds — derivatives, algebra, fractions & more. Includes full answer keys. Perfect for students, teachers & tutors.
⭐ Advanced Mathematics Resources
Explore our most advanced mathematical tools and tutorials designed for university students, researchers, and professionals.
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Advanced tutorial solving the viral math problem: lim ⟨h_p, z_p⟩ = 0.9 and lim ⟨h_p, b_p⟩ = 0.9375. Complete with geometric interpretations, convergence proofs, and interactive visualizations.
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Comprehensive guide to positive definite matrices: tests (Sylvester's criterion), properties, applications in optimization, quadratic forms, and Cholesky decomposition with interactive examples.
Hilbert Spaces Tutorial
Complete introduction to Hilbert spaces: definitions, properties, orthonormal bases, projection theorem, and applications in quantum mechanics and functional analysis with visual examples.
📐 All Differentiation Rules — Reference Guide
This differentiation calculator applies every standard rule automatically. Here is a complete reference of the rules our derivative solver uses, with examples you can click to try instantly.
⚡ Power Rule
d/dx[xⁿ] = n·xⁿ⁻¹
Example: d/dx[x⁵] = 5x⁴
🔗 Chain Rule
d/dx[f(g(x))] = f'(g(x))·g'(x)
Example: d/dx[sin(x²)] = 2x·cos(x²)
✖️ Product Rule
d/dx[u·v] = u'v + uv'
Example: d/dx[x²·eˣ] = 2x·eˣ + x²·eˣ
➗ Quotient Rule
d/dx[u/v] = (u'v − uv') / v²
Example: d/dx[sin(x)/x]
📈 Exponential Rule
d/dx[eˣ] = eˣ | d/dx[aˣ] = aˣ·ln(a)
Example: d/dx[e^(3x)] = 3e^(3x)
🌐 Logarithm Rule
d/dx[ln(x)] = 1/x | d/dx[logₐ(x)] = 1/(x·ln(a))
Example: d/dx[ln(x²+1)]
🎯 Who Uses This Derivative Calculator?
Our free online derivative calculator is used daily by:
High School & University Students
Check homework, understand step-by-step working, and prepare for calculus exams
Teachers & Professors
Generate worked examples, create practice problems, and verify results instantly
Engineers & Scientists
Differentiate complex formulas in physics, fluid dynamics, and control systems
Data Scientists & AI Researchers
Compute gradients, verify backpropagation, and differentiate loss functions
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