# If f(x)= sin3x  and g(x) = -3x , how do you differentiate f(g(x))  using the chain rule?

Jun 30, 2018

$- 9 \cos \left(9 x\right)$

#### Explanation:

Let's plug in $g \left(x\right)$ wherever we see an $x$ in $f \left(x\right)$. This gives us

$f \left(g \left(x\right)\right) = \sin \left(- 9 x\right)$

Now, in our new composite function, we know

$f \left(x\right) = \sin x \implies f ' \left(x\right) = \cos x$

$g \left(x\right) = - 9 x \implies g ' \left(x\right) = - 9$

The Chain Rule says that the derivative of a composite function $f \left(g \left(x\right)\right)$ will be

$f ' \left(g \left(x\right)\right) \cdot g ' \left(x\right)$

Plugging in our values, we will get

$\cos \left(- 9 x\right) \cdot - 9$

$\implies - 9 \cos \left(9 x\right)$

Hope this helps!