d/dxChain Rule Calculator
e.g. x^2, 3x+1
The Chain Rule is a formula for differentiating composite functions — functions of the form h(x) = f(g(x)). It is one of the most important rules in calculus and is essential for differentiating all but the simplest functions. The rule states: h'(x) = f'(g(x)) × g'(x).
- 1If h(x) = f(g(x)), then h'(x) = f'(g(x)) · g'(x)
- 2In Leibniz notation: dy/dx = (dy/du) · (du/dx)
- 3Identify the outer function f and inner function g
- 4Differentiate the outer function with respect to the inner, then multiply by the derivative of the inner
h(x) = sin(x²), outer=sin, inner=x²=h'(x) = cos(x²) · 2xOuter derivative: cos(u); inner derivative: 2x
h(x) = (3x+1)⁵=h'(x) = 5(3x+1)⁴ · 3 = 15(3x+1)⁴
h(x) = e^(x²)=h'(x) = e^(x²) · 2x
h(x) = ln(cos(x))=h'(x) = (1/cos(x)) · (−sin(x)) = −tan(x)
| Outer f(u) | f'(u) | Full Chain Rule Result |
|---|---|---|
| sin(u) | cos(u) | cos(g(x)) · g'(x) |
| cos(u) | −sin(u) | −sin(g(x)) · g'(x) |
| eᵘ | eᵘ | e^(g(x)) · g'(x) |
| ln(u) | 1/u | g'(x) / g(x) |
| uⁿ | nuⁿ⁻¹ | n·g(x)ⁿ⁻¹ · g'(x) |
| √u | 1/(2√u) | g'(x) / (2√g(x)) |
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