Chain Rule Calculator

How to Use This Calculator

1. Enter the outer function g(u) using u as the variable (e.g. sin(u), u^2, exp(u)).
2. Enter the inner function f(x) using x (e.g. x^2, 2*x+1).
3. Enter the point x.
4. Get g'(f(x)), f'(x), their product, and g(f(x)).
5. Use Common Compositions for sin(x²), e^(2x), ln(x²+1) presets with worked steps.

Formula

Example

Frequently Asked Questions

• What is the chain rule? ▼
  The chain rule states that if y = g(f(x)), then dy/dx = g'(f(x))·f'(x). The derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.
• How do I apply the chain rule to sin(x²)? ▼
  Let g(u) = sin(u) and f(x) = x². Then g'(u) = cos(u) and f'(x) = 2x. So d/dx sin(x²) = cos(x²)·2x = 2x·cos(x²).
• What is the triple chain rule? ▼
  For h(g(f(x))): d/dx = h'(g(f(x)))·g'(f(x))·f'(x). Each derivative is evaluated at the output of the function inside it.
• How does the chain rule apply to parametric curves? ▼
  For a parametric curve x(t), y(t): dy/dx = (dy/dt)/(dx/dt), where both numerator and denominator are computed using the chain rule with respect to t.
• What is the multivariable chain rule? ▼
  For z=f(x,y) with x=x(t), y=y(t): dz/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt). The total derivative is the sum of partial derivatives times rates of change.

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