The chain rule handles derivatives of functions inside other functions, which appear everywhere in calculus. Once you learn to spot the outer and inner parts, these problems become straightforward. You can check your work with the free tools on Calc Solver.
What the Chain Rule Says
If a function is built as an outer function wrapped around an inner function, its derivative is the derivative of the outer function times the derivative of the inner function. In symbols, the derivative of f(g(x)) is f'(g(x)) times g'(x). The key step is always identifying which part is outer and which part is inner.
Step 1: Identify the Outer and Inner Functions
Look for the last operation you would do when evaluating the function. In (3x + 1)^5, the power is applied last, so the outer function is the fifth power and the inner function is 3x + 1. In sin(x^2), the sine is applied last, so sine is outer and x^2 is inner.
Step 2: Differentiate the Outer Function
Differentiate the outer function while leaving the inner function untouched. For (3x + 1)^5, the outer derivative is 5(3x + 1)^4. Do not simplify yet, because one more factor is still missing.
Step 3: Multiply by the Derivative of the Inner Function
Multiply by the derivative of the inside. The derivative of 3x + 1 is 3, so the full answer is 5(3x + 1)^4 times 3, which is 15(3x + 1)^4. Forgetting this final multiplication is the most common chain rule mistake.
Worked Example with Trigonometry
Find the derivative of sin(x^2). The outer function is sine, whose derivative is cosine, so start with cos(x^2). The inner function is x^2, whose derivative is 2x. Multiply to get 2x cos(x^2). Check the structure: outer derivative, then inner derivative, nothing left out.
Common Mistakes to Avoid
First, differentiating only the outer function and stopping. Second, mixing up which function is outer and which is inner. Third, applying the chain rule when the power rule alone would do, such as with a plain polynomial. A quick check is to ask what the last evaluation step is.
Practice Problems to Try
Find the derivatives of (2x – 5)^3, cos(4x), and (x^2 + 1)^4. For each one, name the outer and inner functions before differentiating.