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

derivative of cos(ln(x))

The derivative of

\cos(\ln(x))

-\sin(\ln(x))/x

.

Solution. We will use the chain rule to find out what the derivative of

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

is. In other words, we will use the following:

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

where

f(u) = \cos(u)

and

g(x) = \ln(x)

. The derivative of

f(u)

f'(u) = -\sin(u)

, which can be checked here. Also, we see here that

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

. So we get:

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

Substituting the results, we get:

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

Therefore, the derivative of

\cos(\ln(x))

-\sin(\ln(x))/x