Chain Rule

The chain rule is the derivative rule for a function inside another function. If one quantity depends on a middle step, and that middle step depends on $x$, the total rate of change comes from multiplying those two changes together.

What The Chain Rule Says

If $y = f(g(x))$, and $g$ is differentiable at $x$ while $f$ is differentiable at $g(x)$, then:

$$\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$$

In plain language: differentiate the outer function, keep the inner expression in place, then multiply by the derivative of the inner expression.

Intuition

A composite function changes in two layers. First, a small change in $x$ changes the inner expression $g(x)$. Then that change in $g(x)$ changes the outer value $f(g(x))$.

The chain rule connects those layers. It says the overall change is the change from the outside times the change from the inside.

Worked Example

Find the derivative of:

$$y = (3x^2 + 1)^5$$

Here the inner function is:

$$u = 3x^2 + 1$$

and the outer function is:

$$y = u^5$$

Differentiate the outer function first:

$$\frac{dy}{du} = 5u^4$$

Now differentiate the inner function:

$$\frac{du}{dx} = 6x$$

Multiply them:

$$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 5u^4 \cdot 6x$$

Substitute back $u = 3x^2 + 1$:

$$\frac{dy}{dx} = 30x(3x^2 + 1)^4$$

That last factor, $6x$, is the part people most often forget.

Common Mistakes

1. Differentiating the outer function and stopping too early. For $(3x^2 + 1)^5$, $5(3x^2 + 1)^4$ is not the full derivative.
2. Misidentifying the outer function. In $\sin(x^2)$, the outer function is $\sin(\cdot)$, not the square.
3. Using the chain rule when there is no composition. For $x^3 + 1$, you do not need an extra inner derivative.

When You Use It

The chain rule appears whenever functions are nested. Common examples include:

1. Powers of expressions such as $(x^2 + 4x - 1)^7$
2. Trig functions of expressions such as $\sin(5x)$ or $\cos(x^3)$
3. Exponentials and logs such as $e^{x^2}$ or $\ln(1 + x^4)$
4. Implicit differentiation, where several chain rule steps often appear at once

A Fast Check

After differentiating a composite function, ask one question: did the derivative of the inner expression appear somewhere in the answer?

If not, there is a good chance the chain rule step is incomplete.

Try Your Own Version

Take $y = (2x - 3)^4$ and name the inner function before you differentiate. If your final answer does not include the derivative of $2x - 3$, redo the last step and check where it disappeared.