# What is the derivative of f(x) = ln(sinx))?

Apr 22, 2018

$f ' \left(x\right) = \cot x$

#### Explanation:

We'll apply the Chain Rule, which, when applied to logarithms, tells us that if $u$ is some function in terms of $x ,$ then

$\frac{d}{\mathrm{dx}} \ln u = \frac{1}{u} \cdot \frac{\mathrm{du}}{\mathrm{dx}}$

Here, we see $u = \sin x ,$ so

$f ' \left(x\right) = \frac{1}{\sin} x \cdot \frac{d}{\mathrm{dx}} \sin x$

$f ' \left(x\right) = \cos \frac{x}{\sin} x$

$f ' \left(x\right) = \cot x$

Apr 22, 2018

$f ' \left(x\right) = \cot x$

#### Explanation:

$\text{differentiate using the "color(blue)"chain rule}$

$\text{Given "f(x)=g(h(x))" then}$

$f ' \left(x\right) = g ' \left(h \left(x\right)\right) \times h ' \left(x\right) \leftarrow \textcolor{b l u e}{\text{chain rule}}$

$\text{here } f \left(x\right) = \ln \left(\sin x\right)$

$\Rightarrow f ' \left(x\right) = \frac{1}{\sin} x \times \frac{d}{\mathrm{dx}} \left(\sin x\right) = \cos \frac{x}{\sin} x = \cot x$