Chain Rule

The chain rule is a fundamental concept in calculus that allows us to differentiate composite functions. It is an essential tool for calculating the derivatives of complex functions that can be broken down into simpler ones. This blog will delve into the chain rule, explore its formula, provide examples, and discuss its applications in differentiation and integration.

1.0What is the Chain Rule?

In simple terms, the chain rule helps us find the derivative of a function that is composed of two or more functions. If we have a function y = f(g(x)), where g(x) is a function nested inside another function f(x), the chain rule states that:

$\frac{dy}{dx}=\frac{dy}{dg}\cdot \frac{dg}{dx}\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{dy}}{\mathrm{du}}\cdot \frac{\mathrm{du}}{\mathrm{dx}}$

This means that to differentiate y, we first differentiate f with respect to g and then multiply it by the derivative of g with respect to x.

2.0Chain Rule Derivative Formula

The chain rule formula can be expressed in a more generalized way as follows:

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

Where:

• f'(g(x)) is the derivative of the outer function evaluated at the inner function.
• g'(x) is the derivative of the inner function.

3.0Chain Rule Derivative Examples

Let’s look at a couple of examples to illustrate how to apply the chain rule in differentiation:

Example 1: Given $y={\left(3x^2+2\right)}^4$, we can identify $f(u)=u^4$ and $g(x)=3x^2+2.$

1. Find $f^{\prime }(u)=4u^3$ and substitute g(x) into it:  

$f^{\prime }(g(x))=4{\left(3x^2+2\right)}^3$

2. Find g'(x) = 6x.

Now, applying the chain rule:

$\frac{dy}{dx}=f^{\prime }(g(x))\cdot g^{\prime }(x)$

$\frac{dy}{dx}=4{\left(3x^2+2\right)}^3\cdot 6x$

$\frac{dy}{dx}=24x{\left(3x^2+2\right)}^3$

Example 2: Consider $y=\sin \left(2x^3\right)$. Here, we set $f(u)=\sin (u)$ and $g(x)=2x^3$.

1. Find $f^{\prime }(u)=\cos (u)$ and substitute g(x):  

$f^{\prime }(g(x))=\cos \left(2x^3\right)$

2. Find $g^{\prime }(x)=6x^2$ .

Now, applying the chain rule:

$\frac{dy}{dx}=f^{\prime }(g(x))\cdot g^{\prime }(x)$

$\frac{dy}{dx}=\cos \left(2x^3\right)\cdot 6x^2$

$\frac{dy}{dx}=6x^2\cos \left(2x^3\right)$

4.0Chain Rule for Partial Derivatives

The chain rule extends beyond single-variable calculus. In multivariable calculus, we encounter chain rule for partial derivatives, which is useful when dealing with functions of several variables.

If z = f(x, y) and x = g(t), y = h(t), then the total derivative of z with respect to t is given by:

$\frac{dz}{dt}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}$

5.0Chain Rule in Integration

The chain rule in integration, often referred to as the reverse chain rule, is crucial when evaluating integrals involving composite functions. If we have an integral of the form:

$\int f(g(x))\cdot g^{\prime }(x)dx$

we can use the reverse chain rule to simplify our integration process. The result will be:

$\int f(g(x))\cdot g^{\prime }(x)dx=F(g(x))+C$

where F is the antiderivative of f and C is the constant of integration.

6.0Solved Examples on Chain Rule

Example 1: Differentiate $y={\left(2x^3+5\right)}^7$.

Solution:

Given function is:

$y={\left(2x^3+5\right)}^7$

$\text{ Let }y=u^7\text{ and }u=2x^3+5$

$\frac{dy}{du}=7u^6$

$\frac{du}{dx}=\frac{d}{dx}\left(2x^3+5\right)=6x^2$

Now, using the chain rule, we have.

$\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}$

$\frac{dy}{dx}=7u^6\cdot 6x^2$

$\frac{dy}{dx}=7{\left(2x^3+5\right)}^6\cdot 6x^2$

$\frac{dy}{dx}=42x^2{\left(2x^3+5\right)}^6$

Thus, the derivative of $y={\left(2x^3+5\right)}^7\text{ is }\frac{dy}{dx}=42x^2{\left(2x^3+5\right)}^6.$

Example 2: Find the derivative of y=\sin \left(4 x^2+3\right) .

Solution:

Given function is:

$y=\sin \left(4x^2+3\right)$

$\text{ Let }y=\sin (u)\text{ and }u=4x^2+3$

$\frac{dy}{du}=\cos (u)$

$\frac{du}{dx}=\frac{d}{dx}\left(4x^2+3\right)=8x$

Now, using the chain rule, we have.

$\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}$

$\frac{dy}{dx}=\cos (u)\cdot 8x$

$\frac{dy}{dx}=8x\cos \left(4x^2+3\right)$

Thus, the derivative of

$y=\sin \left(4x^2+3\right)\text{ is }\frac{dy}{dx}=8x\cos \left(4x^2+3\right)$.

Example 3: Differentiate $y=e^{\ln \left(x^2+1\right)}$.

Solution:

Given function is:

$y=e^{\ln \left(x^2+1\right)}$

$\text{ Let }y=e^u\text{ and }u=\ln \left(x^2+1\right)$.

$\frac{dy}{du}=e^u$

$\frac{du}{dx}=\frac{d}{dx}\left(\ln \left(x^2+1\right)\right)$

$\frac{du}{dx}=\frac{1}{x^2+1}\cdot \frac{d}{dx}\left(x^2+1\right)$

$\frac{du}{dx}=\frac{2x}{x^2+1}$

Now, using the chain rule, we have.

$\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}$

$\frac{dy}{dx}=e^u\cdot \frac{2x}{x^2+1}$

$\frac{dy}{dx}=e^{\ln \left(x^2+1\right)}\cdot \frac{2x}{x^2+1}$

$\frac{dy}{dx}=\left(x^2+1\right)\cdot \frac{2x}{x^2+1}=2x$

Thus, the derivative of

$y=e^{\ln \left(x^2+1\right)}\text{ is }\frac{dy}{dx}=\left(x^2+1\right)\cdot \frac{2x}{x^2+1}=2x.$

Example 4: If $z=f(x,y)=x^2+y^2\text{ and }\mathrm{x}=2\mathrm{t},\mathrm{y}=3\mathrm{t}\text{, find }\frac{dz}{dt}$.

Solution:

1. Find the partial derivatives of z:  

$\frac{\partial z}{\partial x}=2x$

$\frac{\partial z}{\partial y}=2y$

2. Find $\frac{dx}{dt}\text{ and }\frac{dy}{dt}:$

$\frac{dx}{dt}=2$

$\frac{dy}{dt}=3$

3. Apply the chain rule for partial derivatives:  

$\frac{dz}{dt}=\frac{\partial z}{\partial x}\cdot \frac{dx}{dt}+\frac{\partial z}{\partial y}\cdot \frac{dy}{dt}$

$\frac{dz}{dt}=2x\cdot 2+2y\cdot 3$

4. Substitute x = 2t and y = 3t:  

$\frac{dz}{dt}=2(2t)\cdot 2+2(3t)\cdot 3=8t+18t=26t$

7.0Practice Questions on Chain Rule

1. Differentiate the following functions using the chain rule:

• $y={\left(5x^3-2\right)}^4$
• $y=e^{3x^2}$
• $y=\sin (4x)$
• $y=\ln \left(x^2+1\right)$

2. Find the derivative of $y=(2x+5{)}^6$  using the chain rule.
3. Determine $\frac{dy}{dx}\text{ for }y=\tan \left(3x^2+2\right).$
4. Differentiate $y={\left(x^2+1\right)}^3+{\left(2x^2+3\right)}^4$ .
5. Find the derivative of $y=\cos (2x-5{)}^2$ .

8.0Sample Questions on Chain Rule

1. What is the chain rule in calculus? 

Ans: The chain rule is a method used to differentiate composite functions. If a function y = f(g(x)) is composed of two functions f and g, the chain rule states that:

$\frac{dy}{dx}=f^{\prime }(g(x))\cdot g^{\prime }(x)$

This formula helps in finding the derivative of a function that is nested within another function.

2. How do I know when to use the chain rule?

Ans: Use the chain rule when you encounter a composite function—that is, a function where one function is inside another. For instance, if you see expressions like ${\left(3x^2+2\right)}^5,\sin \left(4x^2\right),\text{ or }{e}^{x^3}$, the chain rule is the appropriate technique for differentiation.

3. How does the chain rule work for more than two functions?

Ans: The chain rule can be extended to differentiate a function composed of more than two functions. If y = f(g(h(x))), where f, g, and h are three functions, the chain rule is applied as:

$\frac{dy}{dx}=f^{\prime }(g(h(x)))\cdot g^{\prime }(h(x))\cdot h^{\prime }(x)$

This formula can be extended to as many layers of functions as necessary.

4. Can the chain rule be applied to partial derivatives?

Ans: Yes, the chain rule for partial derivatives is used when differentiating a function of multiple variables. If z = f(x, y), and x and y are functions of another variable t, then the chain rule for partial derivatives is given by:

$\frac{dz}{dt}=\frac{\partial z}{\partial x}\frac{dx}{dt}+\frac{\partial z}{\partial y}\frac{dy}{dt}$

This rule helps find how z changes with t through its dependencies on x and y.

5. Can the chain rule be used for trigonometric functions?

Ans: Yes, the chain rule is highly effective for differentiating trigonometric functions that are composed with other functions. For example, if $y=\sin (2x)$ , the derivative using the chain rule is:

$\frac{dy}{dx}=\cos (2x)\cdot 2=2\cos (2x)$

Similarly, the chain rule can be used for functions involving cos(x), tan(x), and other trigonometric forms.