Chain Rule of Derivatives: Statement, Formula, Proof, Examples

The chain rule of derivatives is used to find the derivative of a composite function. The rule states that the derivative of a composite function f(g(x)) is equal to f'(g(x)) ⋅ g'(x). In this post, we will learn the statement of the chain rule, its proof, step-by-step method to use this rule along with solve examples.

Statement of Chain Rule

Let f(x) and g(x) be two functions of x. Then the derivative of the composite function f(g(x)) is computed by the chain rule and is equal to

$[f(g(x))]’ = f'(g(x)) \cdot g'(x)$.

Proof of Chain Rule Formula

By the first principle, i.e, by definition the derivative of f(g(x)) is given by the following limit

[f(g(x))]’ = lim_h→0 $\dfrac{f(g(x+h)) – f(g(x))}{h}$

= lim_h→0 $\Big[ \dfrac{f(g(x+h)) -f(g(x))}{g(x+h)-g(x)}$ $\times \dfrac{g(x+h) – g(x)}{h} \Big]$

= lim_h→0 $\dfrac{f(g(x+h)) -f(g(x))}{g(x+h)-g(x)}$ × lim_h→0$\dfrac{g(x+h) – g(x)}{h}$

= f'(g(x)) ⋅ g'(x), obtained by the definition of derivatives at a point.

Thus, we have shown that [f(g(x))]’ = f'(g(x)) ⋅ g'(x) and this is the chain rule of derivatives. ♣

Chain Rule Step-by-Step

The following steps are used in order to find the derivative of a composite function y(x) using chain rule:

Step 1: First check that y(x) is a composite function or not.

Step 2: If y(x) is composite, then it can be written as f(g(x)) where g(x) is the inner function and f(x) is the outer function.

Step 3: Now, determine the inner and outer functions.

Step 4: Find the derivatives of the inner and outer functions.

Step 5: Compute f'(g(x)) ⋅ g'(x).

Step 6: Simplify the above product.

Step 7: The simplification in the previous step is the derivative of the composite function y(x) =f(g(x)).

Before we provide examples on how to find the derivatives by chain rule, let us recall its formula.

Chain Rule of Derivatives Formula: Let y(x) = f(g(x)) be a composite function. Then the chain rule formula for the derivatives of y(x) is given by $\dfrac{dy}{dx}$ = $\dfrac{dy}{du} \cdot \dfrac{du}{dx}$ where u=g(x), so y=f(u).

Solved Examples on Chain Rule

Example 1: Find the derivative of e3x.

Let us find the derivative of e^3x by the chain rule as it is a composite function. Note that we can write it as

f(g(x)) where g(x)=3x and f(x)=e^x.

Let u=3x. So du/dx =3.

Now, by the chain rule,

d/dx (e^3x)

= d/du (e^u) ⋅ du/dx

= e^u ⋅ 3

= 3e^3x as u=3x.

So the derivative of e^3x by the chain rule is 3e^3x.

Example 2: Find the derivative of sin(2x).

Note that sin(2x) can be written as f(g(x)) where g(x)=2x and f(x)=sin(x).

Put u=2x. So du/dx =2.

By the chain rule of derivatives,

d/dx (sin2x)

= d/du (sin u) ⋅ du/dx

= cos(u) ⋅ 2 since the derivative of sinx is cosx.

= 2cos(2x) as u=2x.

So the derivative of sin(2x) by the chain rule is 2cos(2x).

FAQs on Chain Rule