What is the derivative of sin(ln(x))?

derivative of sin(ln(x))

The derivative of

\sin(\ln(x))

\cos(\ln(x))/x

.

Solution. To determine the derivative

F(x) = f(g(x)) = \sin(\ln(x))

, we will use the chain rule:

\begin{align*} F'(x) = f'(g(x))g'(x), \end{align*}

where

f(u) = \sin(u)

and

g(x) = \ln(x)

. We have seen here

f'(u) = \cos(u)

and here that

g'(x) = \frac{1}{x}

. So we get:

\begin{align*} f'(g(x)) = \cos(g(x)) = \cos(\ln(x)). \end{align*}

Combining everything, we get:

\begin{align*} F'(x) &= f'(g(x))g'(x) \\ &= \cos(\ln(x))\frac{1}{x} \\ &= \frac{\cos(\ln(x))}{x}. \end{align*}

Therefore, the derivative of

\sin(\ln(x))

\cos(\ln(x))/x