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## The Chain Rule

The chain rule allows you to differentiate composite functions (functions of other functions) ie) f(g(x)) such as sin(3x2) or (5x3+2x+3)2. The rule is as follows
$\frac{d}{dx}(f(g(x)) = \frac{dg}{dx}\frac{df}{dg}(g(x)) = g'(x)f'(g(x))$
or to understand it more simply you differentiate the inner function and multiply it by the derivative of the outer function (leaving what’s inside alone).

## Differentiating brackets raised to a power

The chain rule can be a great short cut to differentiating brackets raised to a power as it doesn’t require you to multiply them all out, it also enables you to differentiate brackets raised to an unknown power.
Consider $\frac{d}{dx}((ax + b)^n)$
This is the composite of the functions ax+b and tn. So we differentiate them both to get a and ntn-1 and then apply the formula to get
$\frac{d}{dx}((ax + b)^n) = an(ax+b)^{n-1}$
Notice how we multiplied the derivative of the inner function, a, by the derivative of the outer function ntn-1 but substituted ax+b back in for t.

To generalise we can replace the ax+b with f(x) and by applying the above get
$\frac{d}{dx}((f(x))^n) = f'(x)n(f(x))^{n-1}$

## Differentiating Trigonometric functions

We can also use the chain rule when differentiating sin(f(x)) and cos(f(x)) since we know how to differentiate sin(x) and cos(x).
Using the chain rule we get
$\frac{d}{dx}(sin(f(x)) = f'(x)cos(f(x))$
and
$\frac{d}{dx}(cos(f(x)) = -f'(x)sin(f(x))$