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Free chain rule calculator & derivative calculator. Calculate chain rule derivatives, composite functions & implicit differentiation with step-by-step solutions. Our calculator uses the chain rule formula to find derivatives of nested functions including trigonometric, exponential, and polynomial compositions.
Last updated: February 2, 2026
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Select the outer function (the function applied last)
Select the inner function (the function applied first)
Composite Function:
f(g(x)) = sin((x²))
Enter a value to evaluate the derivative at that point
Outer Function Derivative:
f'(u) = cos(u)
Inner Function Derivative:
g'(x) = 2x
Chain Rule Formula:
dy/dx = f'(g(x)) × g'(x)
Final Derivative:
dy/dx = (cos(u)) × (2x)
Solution Steps:
Chain Rule Tips:
Formula
dy/dx = f'(g(x)) × g'(x)
Find derivatives of nested functions using the chain rule formula
Composition
f(g(x)) = (f ∘ g)(x)
Break down composite functions into outer and inner components
Example
d/dx[sin(x²)] = 2x cos(x²)
Handle sine, cosine, and tangent with inner functions
Example
d/dx[e^(3x)] = 3e^(3x)
Differentiate exponential functions with variable exponents
Technique
d/dx[y²] = 2y dy/dx
Apply chain rule when y is a function of x implicitly
Extended form
f'(g(h(x))) × g'(h(x)) × h'(x)
Handle deeply nested composite functions
Find the derivative of y = sin(x²) using the chain rule
Outer derivative
f'(u) = cos(u)
Inner derivative
g'(x) = 2x
Chain Rule Result
dy/dx = 2x cos(x²)
Our chain rule calculator finds derivatives of composite functions by applying the fundamental chain rule formula from calculus. The calculator identifies the outer and inner functions, computes their individual derivatives, and multiplies them together to get the final result.
If y = f(g(x)), then:
dy/dx = f'(g(x)) × g'(x)or equivalently:
dy/dx = dy/du × du/dxThe chain rule tells us that to differentiate a composite function, we take the derivative of the outer function (evaluated at the inner function) and multiply by the derivative of the inner function. This is one of the most important and frequently used rules in differential calculus.
Shows the flow from composite function to derivative using chain rule
The chain rule is based on the concept of composite function differentiation. When a function y depends on u, and u depends on x, then y indirectly depends on x. The rate of change of y with respect to x is the product of the rate of change of y with respect to u and the rate of change of u with respect to x. This multiplicative relationship is the essence of the chain rule.
Need help with other derivative calculations? Check out our derivative calculator and implicit derivative calculator.
Get Custom Calculator for Your PlatformFinal Answer: dy/dx = 2x cos(x²)
The derivative of sin(x²) is the derivative of sine (which is cosine) times the derivative of x² (which is 2x).
y = e^(3x)
dy/dx = e^(3x) × 3 = 3e^(3x)
y = (2x + 1)³
dy/dx = 3(2x + 1)² × 2 = 6(2x + 1)²
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