⛓️ Chain Rule Calculator
Calculate derivatives of composite functions with step-by-step solutions and detailed explanations
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What is the Chain Rule?
The chain rule is one of the most important differentiation rules in calculus. It's used to find the derivative of composite functions—functions that are "nested" inside other functions.
In simple terms: "Derivative of the outer function times the derivative of the inner function."
When to Use the Chain Rule
Use the chain rule whenever you have a composite function, which means one function is inside another. Common examples include:
- Exponential composites: e^(3x+1), e^(x²)
- Trigonometric composites: sin(x²), cos(3x), tan(e^x)
- Powers of functions: (x²+1)⁵, (sin(x))³
- Logarithmic composites: ln(x³), log(2x+5)
- Square roots: √(x²+4), √(sin(x))
How the Chain Rule Works
The chain rule works by breaking down the composite function into two parts:
- Outer function (f): The outside operation
- Inner function (g): The inside operation
Example 1: Find d/dx[sin(x²)]
Step 1: Identify outer and inner functions
- Outer: f(u) = sin(u)
- Inner: g(x) = x²
Step 2: Find derivatives
- f'(u) = cos(u)
- g'(x) = 2x
Step 3: Apply chain rule
d/dx[sin(x²)] = cos(x²) · 2x = 2x·cos(x²)
Example 2: Find d/dx[e^(3x+1)]
Step 1: Identify functions
- Outer: f(u) = e^u
- Inner: g(x) = 3x+1
Step 2: Find derivatives
Step 3: Apply chain rule
d/dx[e^(3x+1)] = e^(3x+1) · 3 = 3e^(3x+1)
Common Chain Rule Patterns
- d/dx[sin(u)] = cos(u) · u'
- d/dx[cos(u)] = -sin(u) · u'
- d/dx[e^u] = e^u · u'
- d/dx[ln(u)] = (1/u) · u'
- d/dx[u^n] = n·u^(n-1) · u'
Tips for Mastering the Chain Rule
- Always identify the outer and inner functions first
- Remember: "Outside times inside" - differentiate the outside, then multiply by the derivative of the inside
- Practice with simple examples before moving to complex ones
- Check your work by expanding if possible
- Use parentheses to keep track of what you're differentiating