Derivative of sin(lnx)

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sin(log(x))

$$\sin{\left(\log{\left(x \right)} \right)}$$

sin(log(x))

Detail solution

Let $u = \log{\left(x \right)}$ .
The derivative of sine is cosine:

$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
The result of the chain rule is:

$\frac{\cos{\left(\log{\left(x \right)} \right)}}{x}$

The answer is:

$\frac{\cos{\left(\log{\left(x \right)} \right)}}{x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

cos(log(x))
-----------
     x

$$\frac{\cos{\left(\log{\left(x \right)} \right)}}{x}$$

Simplify

The second derivative [src]

-(cos(log(x)) + sin(log(x))) 
-----------------------------
               2             
              x

$$- \frac{\sin{\left(\log{\left(x \right)} \right)} + \cos{\left(\log{\left(x \right)} \right)}}{x^{2}}$$

Simplify

The third derivative [src]

3*sin(log(x)) + cos(log(x))
---------------------------
              3            
             x

$$\frac{3 \sin{\left(\log{\left(x \right)} \right)} + \cos{\left(\log{\left(x \right)} \right)}}{x^{3}}$$

Simplify

The graph