Derivative FAQ

A derivative measures how a function changes as its input changes. It represents the rate of change or slope of a function at any point.

Geometrically, the derivative is the slope of the tangent line to the function's graph. Algebraically, it's defined as:

f'(x) = lim(h→0) [f(x+h) - f(x)]/h

Learn more: What is a Derivative?

Derivatives are essential for:

• Physics: Calculating velocity, acceleration, and force
• Engineering: Optimizing designs and analyzing systems
• Economics: Finding marginal cost and revenue
• Biology: Modeling population growth and decay
• Everyday life: From speedometers to GPS navigation

See real-world examples: Applications of Derivatives

They represent the same thing - the derivative of f with respect to x. Different notations are used in different contexts:

• f'(x) - Lagrange notation (prime notation)
• df/dx - Leibniz notation
• Df - Euler notation
• ẋ - Newton notation (for time derivatives)

The power rule is the most fundamental derivative rule:

d/dx[x^n] = n·x^(n-1)

Examples:

• d/dx[x³] = 3x²
• d/dx[x⁵] = 5x⁴
• d/dx[√x] = d/dx[x^(1/2)] = (1/2)x^(-1/2)

Complete reference: All Derivative Formulas

Use the chain rule when differentiating composite functions (functions inside functions):

d/dx[f(g(x))] = f'(g(x))·g'(x)

Examples:

• d/dx[sin(x²)] = cos(x²)·2x
• d/dx[(3x + 1)⁴] = 4(3x + 1)³·3
• d/dx[e^(x²)] = e^(x²)·2x

Master the technique: Chain Rule Complete Guide

Use the product rule when differentiating the product of two functions:

(uv)' = u'v + uv'

Example: d/dx[x²·sin(x)] = 2x·sin(x) + x²·cos(x)

Memory aid: "First times derivative of second, plus second times derivative of first"

Learn more: Product Rule Tutorial

The quotient rule is for differentiating fractions:

(u/v)' = (u'v - uv')/v²

Memory aid: "Low d-high minus high d-low, over the square of what's below"

Example: d/dx[(x² + 1)/(x - 2)] = [(2x)(x - 2) - (x² + 1)(1)]/(x - 2)²

Yes! You can rewrite division as multiplication and use the product rule:

u/v = u·v^(-1)

Then apply product rule with chain rule. However, quotient rule is usually faster and more direct.

d/dx[sin(x)] = cos(x)

All trig derivatives:

• d/dx[sin(x)] = cos(x)
• d/dx[cos(x)] = -sin(x)
• d/dx[tan(x)] = sec²(x)

Full list: Trigonometric Formulas

d/dx[e^x] = e^x

The exponential function e^x is unique - it's the only function that equals its own derivative!

For other exponential bases: d/dx[a^x] = a^x·ln(a)

d/dx[ln(x)] = 1/x

For any logarithmic base: d/dx[log_a(x)] = 1/(x·ln(a))

Important: This only works for x > 0. For all x ≠ 0, use ln|x|.

d/dx[c] = 0

The derivative of any constant is zero because constants don't change!

Examples: d/dx[5] = 0, d/dx[π] = 0, d/dx[100] = 0

Implicit differentiation is used when the equation is not solved for y (e.g., x² + y² = 25).

Steps:

1. Differentiate both sides with respect to x
2. Remember: d/dx[y] = dy/dx (use chain rule)
3. Solve for dy/dx

Complete guide: Implicit Differentiation

Higher-order derivatives are derivatives of derivatives:

• First derivative f'(x): Rate of change, slope
• Second derivative f''(x): Concavity, acceleration
• Third derivative f'''(x): Jerk (rate of change of acceleration)

Example: If f(x) = x⁴, then f'(x) = 4x³, f''(x) = 12x², f'''(x) = 24x

Related rates problems involve finding how fast one quantity is changing based on how fast another quantity is changing.

Example: A ladder is sliding down a wall. If the bottom slides away at 2 ft/s, how fast is the top sliding down?

Strategy:

1. Draw a diagram
2. Write an equation relating the variables
3. Differentiate both sides with respect to time
4. Plug in known values and solve

Ask yourself these questions in order:

1. Is it a basic function? Use memorized derivatives
2. Is it a sum/difference? Differentiate term by term
3. Is it a product? Use product rule
4. Is it a quotient? Use quotient rule
5. Is it a composition (function inside function)? Use chain rule
6. Not solved for y? Use implicit differentiation

Practice identifying: Practice Problems

Top mistakes to avoid:

1. Product rule error: (uv)' ≠ u'v' (must use u'v + uv')
2. Forgetting chain rule: d/dx[sin(x²)] ≠ cos(x²)
3. Sign errors in trig: d/dx[cos(x)] = -sin(x) (note the negative!)
4. Quotient rule order: Use (u'v - uv')/v², not (uv' - u'v)/v²
5. Treating constants as variables

Verification methods:

• Use our calculator: Derivative Calculator with step-by-step solutions
• Check units: Derivative units should match (distance/time → velocity)
• Test with simple values: Plug in x = 0 or x = 1
• Graph both: The derivative should match the slope
• Differentiate again: Second derivative can reveal errors

We offer multiple practice resources:

• Interactive Practice Problems - 30+ problems with instant solutions
• Printable Worksheets - 20+ PDFs with answer keys
• Solved Examples - Step-by-step walkthroughs
• Video Tutorials - Visual explanations

Steps:

1. Go to Derivative Calculator
2. Enter your function (e.g., x^2, sin(x), e^x)
3. Click "Calculate"
4. View step-by-step solution

The calculator shows which rules were applied and provides complete working!

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