Chain Rule Examples

Example 1:
Differentiate $y = {\left( {2{x^3} - 5{x^2} + 4} \right)^5}$ with respect to $x$ using chain rule method.

Let us consider $u = 2{x^3} - 5{x^2} + 4$, then $y = {u^5}$. Applying chain rule, we have

Example 2:
Differentiate $y = {x^2} + 4$ with respect to $\sqrt {{x^2} + 1}$ using chain rule method.

If $u = \sqrt {{x^2} + 1}$, then we have to find $\frac{{dy}}{{du}}$. By chain rule method

First we differentiate the function $y = {x^2} + 4$ with respect to $x$.

Now differentiate the function $u = \sqrt {{x^2} + 1}$ with respect to $x$.

Now using the chain rule of differentiation, we have