Chain Rule Derivative Calculator

The chain rule is a differentiation rule used to find the derivative of a composite function — a function of a function. If f(x) = g(h(x)), then f′(x) = g′(h(x)) · h′(x). You differentiate the outer function (leaving the inner function intact), then multiply by the derivative of the inner function.

A composite function is one where you evaluate a function inside another function. For example, f(x) = sin(x²) is composite because you first compute x², then apply sin to the result. The chain rule is the correct differentiation technique for any such function.

First, identify the outer function g(u) and the inner function u = h(x). Second, differentiate the outer function with respect to u, giving g′(u). Third, differentiate the inner function with respect to x, giving h′(x). Finally, multiply: f′(x) = g′(h(x)) · h′(x), substituting h(x) back in for u.

Yes. When a composite function is nested multiple levels deep — such as sin(cos(x²)) — you apply the chain rule repeatedly, working from the outermost layer inward. This calculator supports first and second-order differentiation for common composite forms.

The chain rule applies when one function is composed inside another, f(g(x)). The product rule applies when two separate functions are multiplied together, f(x)·g(x). Many expressions require both rules at once, but for pure composite functions, the chain rule alone is sufficient.

The chain rule is used throughout physics, engineering, economics, and machine learning. Key applications include differentiating trigonometric, exponential, and logarithmic functions with non-trivial arguments, computing gradients in neural networks (backpropagation), and solving related rates problems in physics and geometry.

For multivariable functions, the partial derivative chain rule extends the single-variable rule. If z = f(x, y) and both x and y depend on a parameter t, then dz/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt). This is critical in thermodynamics, fluid dynamics, and optimization.

The chain rule is one of the most frequently used rules in calculus. Almost every real-world function is composite in nature, so mastering the chain rule unlocks the ability to differentiate a vast range of functions. It is foundational for studying integrals via substitution, differential equations, and multivariable calculus.