All off the following expressions $(x^{3}+5)^{2}$, $e^{x^{2}}$, $\sin (x^{3})$ are the example of composite functions. To evaluate them for a particular value of x, you would need to work out the come off one function, and then use a second function to arrive at the final answer.

**Example,** for the expression $(x^{3}+5)^{2}$

$y$ = $(x^{3}+5)^{2}$ then the function can be broken down into two steps:

$u$ = $x^{3}+5$ and $y$ = $u^{2}$

The alternative way of writing the chain rule helps to understand the concept well,

If $y$ = $f(g(x))$ then $y$ = $f(u)$ and $u$ = $g(x)$

so** $\frac{dy}{dx}$ = $\frac{dy}{du}$ $\times$ ****$\frac{du}{dx}$**

= $f '(u) \times g'(x)$

= $g'(x).f'(g(x))$

Solution:

Here the derivative of ($x^{2}$+3x) is (2x+3).

Let $y$ = $(x^{2}+3x)^{4}$

We can split the above expression into

$y$ = $u^{4}$ and $u$ = $x^{2}+3x$

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

$\frac{du}{dx}$ = $2x+3$

By the chain rule,

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

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

By reversing the above expression, we have

$\int 4(x^{2}+3x)^{3}(2x+3)dx$ = $(x^{2}+3x)^{4}+c$

**Integration by parts: **

$\int u\ dv$ = $uv - \int v\ du$