# How do you find the derivative of the function f(w) = ln(sin(w−15))?

Mar 5, 2018

cot(w-15))

#### Explanation:

this is just by using the Chain rule
https://en.wikipedia.org/wiki/Chain_rule

the derivative of the inside function $\sin \left(w - 15\right)$ is $\cos \left(w - 15\right)$
and the derivative of the outside function $\frac{d}{\mathrm{dx}} \ln \left(x\right) = \frac{1}{x}$

therefore, the derivative of this whole is the derivative of the outside function evaluated at the inside function multiplied by the derivative of the inside function
that is,
$\frac{d}{\mathrm{dx}} f \left(g \left(x\right)\right) = f ' \left(g \left(x\right)\right) \cdot g ' \left(x\right)$

that is equal to
$\frac{1}{\sin} \left(w - 15\right) \cdot \cos \left(w - 15\right)$

which is
$\cos \frac{w - 15}{\sin} \left(w - 15\right)$

and that is equal to

.