Derivatives Evaluate the following derivatives. ddx(sin⁡(ln⁡x))

Question

Derivatives Evaluate the following derivatives.  ddx(sin⁡(ln⁡x))

alkaholikd9 · Accepted Answer

Step 1
Derivative of a given function is to be calculated.
The given function is

\sin (\ln x)

.
The derivative of

\sin x

and

\ln x

is

\cos x

and

\frac{1}{x}

respectively.
Step 2
Now apply chain rule to get the desired derivative,

\frac{d}{d x} (\sin (\ln x)) = \cos (\ln x) \cdot \frac{1}{x}

= \frac{\cos (\ln x)}{x}

Hence the derivative

\frac{d}{d x} (\sin (\ln x))

is

\frac{\cos (\ln x)}{x}

.

Steve Hirano · Accepted Answer

We need to find the derivative function

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

f^{'} (x) = \frac{d}{d x} (f (x))

= \frac{d}{d x} (\sin (\ln x))

Use composite function rule of differentiation

\frac{d}{d x} (f (g (x)) = f^{'} (g (x) \cdot g^{'} (x)

Use differentiation formula

\frac{d}{d x} \sin x = \cos x

f^{'} (x) = \cos (\ln x) \cdot \frac{d}{d x} \ln x

Use differentiation formula

\frac{d}{d x} \ln x = \frac{1}{x}

f^{'} (x) = \cos (\ln x) \cdot \frac{1}{x}

f^{'} (x) = \frac{\cos (\ln x)}{x}

Result:

f^{'} (x) = \frac{\cos (\ln x)}{x}

Derivatives Evaluate the following derivatives. \frac{d}{dx}(\sin (\ln x))