Derivative of sinln5x

You have entered [src]

sin(log(5*x))

$$\sin{\left(\log{\left(5 x \right)} \right)}$$

sin(log(5*x))

Detail solution

Let $u = \log{\left(5 x \right)}$ .
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 $\frac{d}{d x} \log{\left(5 x \right)}$ :
1. Let $u = 5 x$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $5$
  The result of the chain rule is:
  
  $\frac{1}{x}$
The result of the chain rule is:

$\frac{\cos{\left(\log{\left(5 x \right)} \right)}}{x}$

The answer is:

$\frac{\cos{\left(\log{\left(5 x \right)} \right)}}{x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

cos(log(5*x))
-------------
      x

$$\frac{\cos{\left(\log{\left(5 x \right)} \right)}}{x}$$

Simplify

The second derivative [src]

-(cos(log(5*x)) + sin(log(5*x))) 
---------------------------------
                 2               
                x

$$- \frac{\sin{\left(\log{\left(5 x \right)} \right)} + \cos{\left(\log{\left(5 x \right)} \right)}}{x^{2}}$$

Simplify

The third derivative [src]

3*sin(log(5*x)) + cos(log(5*x))
-------------------------------
                3              
               x

$$\frac{3 \sin{\left(\log{\left(5 x \right)} \right)} + \cos{\left(\log{\left(5 x \right)} \right)}}{x^{3}}$$

Simplify