The Chain Rule Explained (With Examples)

The Power, Sum, Product, and Quotient rules give you a massive arsenal for tackling calculus problems. But eventually, you will run into a function that doesn’t fit any of those molds. What do you do when a function is stuffed inside another function?

Think of equations like $y = \sin(x^2)$ or $y = (3x + 1)^5$ . You cannot use the Power Rule alone, and it isn’t two separate functions being multiplied. It is a composite function—a mathematical Russian nesting doll.

To find the derivative of these nested functions, you need The Chain Rule. It is arguably the most frequently used rule in all of calculus. Here is exactly how it works, the easiest way to visualize it, and several step-by-step examples.

What is the Chain Rule?

The Chain Rule tells us how to differentiate composite functions. The golden rule is simple: Take the derivative of the outside function (leaving the inside exactly as it is), and multiply it by the derivative of the inside function.

Mathematically, if you have a composite function $h(x) = f(g(x))$ , its derivative is:
$h'(x) = f'(g(x)) \cdot g'(x)$
In Leibniz notation, the “chain” concept is even clearer. It looks like interlocking gears transferring a rate of change:
$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$
If gear A turns gear B, and gear B turns gear C, you multiply their individual rates of change to find out how fast C is turning relative to A. You just work your way from the outside layer to the inside layer, multiplying as you go.

5 Examples of the Chain Rule in Action

Let’s unbox some composite functions, starting from a basic polynomial and moving to a complex “double chain.”

Example 1: The “Power Chain” Rule

Let’s find the derivative of a binomial raised to a power:

$f(x) = (2x + 1)^3$

Outside function: Something raised to the 3rd power, $(\dots)^3$ .
Inside function: $2x + 1$ .

Step 1: Take the derivative of the outside using the Power Rule, leaving the inside alone.

$3(2x + 1)^2$
Step 2: Multiply by the derivative of the inside. The derivative of $2x + 1$ is simply $2$ .

$f'(x) = 3(2x + 1)^2 \cdot (2)$
Step 3: Simplify.

$f'(x) = 6(2x + 1)^2$

Example 2: Trigonometry and the Chain Rule

Trig functions almost always require the chain rule unless the inside angle is just a plain $x$ .

$f(x) = \sin(4x^2)$

Outside function: $\sin(\dots)$ .
Inside function: $4x^2$ .

Step 1: Derivative of the outside (derivative of sine is cosine), keeping the inside untouched.

$\cos(4x^2)$
Step 2: Multiply by the derivative of the inside ( $4x^2$ becomes $8x$ ).

$f'(x) = \cos(4x^2) \cdot 8x$
Rewrite it cleanly:

$f'(x) = 8x \cos(4x^2)$

Example 3: Exponentials with Complex Powers

The derivative of $e^x$ is $e^x$ , but what if the exponent is an entire equation?

$f(x) = e^{5x - 3}$

Outside function: $e^{(\dots)}$ .
Inside function: $5x - 3$ .

Step 1: The derivative of the outside $e^{(\dots)}$ is just itself.

$e^{5x - 3}$
Step 2: Multiply by the derivative of the inside exponent.

$f'(x) = e^{5x - 3} \cdot (5)$
$f'(x) = 5e^{5x - 3}$

Example 4: Dealing with Square Roots

Radicals are just fractional exponents in disguise, meaning they are perfect candidates for the Chain Rule.

$f(x) = \sqrt{x^3 + 2}$
First, rewrite it with a fractional exponent so you can see the layers clearly:

$f(x) = (x^3 + 2)^{1/2}$
Step 1: Derivative of the outside (Power Rule). Bring down the $1/2$ and subtract 1 from the exponent.

$\frac{1}{2}(x^3 + 2)^{-1/2}$
Step 2: Multiply by the derivative of the inside ( $x^3 + 2$ becomes $3x^2$ ).

$f'(x) = \frac{1}{2}(x^3 + 2)^{-1/2} \cdot (3x^2)$
Step 3: Clean up the negative exponent by pushing it to the denominator.

$f'(x) = \frac{3x^2}{2\sqrt{x^3 + 2}}$

Example 5: The “Double Chain” (Three Layers Deep)

What happens if you have a function inside a function inside a function? You just keep unboxing.

$f(x) = \sin^3(4x)$
First, rewrite it so the layers are painfully obvious:

$f(x) = [\sin(4x)]^3$

Outside layer: $[\dots]^3$
Middle layer: $\sin(\dots)$
Inside layer: $4x$

Step 1: Derivative of the outermost layer.

$3[\sin(4x)]^2$
Step 2: Multiply by the derivative of the middle layer (derivative of $\sin(\dots)$ is $\cos(\dots)$ ).

$3[\sin(4x)]^2 \cdot \cos(4x)$
Step 3: Multiply by the derivative of the innermost layer (derivative of $4x$ is $4$ ).

$f'(x) = 3[\sin(4x)]^2 \cdot \cos(4x) \cdot 4$
Step 4: Combine the constants to simplify the final answer.

$f'(x) = 12\sin^2(4x)\cos(4x)$

Why This Rule Matters

The Chain Rule is the glue that holds differential calculus together. Very few real-world systems operate in isolation. In physics, an object’s temperature might depend on its volume, and its volume might depend on the pressure, which ultimately depends on time.

By allowing us to mathematically unbox interconnected systems one layer at a time, the Chain Rule lets us calculate complex, cascading rates of change without losing our minds. Just remember to work from the outside in, and never change the inside until it is that layer’s turn!

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The Chain Rule Explained (With Examples)

What is the Chain Rule?