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Identical expressions
log((sin(x))^ four)/log(five)
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Similar expressions
log((sinx)^4)/log(5)

The derivative of the function/ log((sin(x))^4)/log(5)

Derivative of log((sin(x))^4)/log(5)

The solution

You have entered [src]

   /   4   \
log\sin (x)/
------------
   log(5)

$$\frac{\log{\left(\sin^{4}{\left(x \right)} \right)}}{\log{\left(5 \right)}}$$

log(sin(x)^4)/log(5)

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \sin^{4}{\left(x \right)}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin^{4}{\left(x \right)}$ :
  1. Let $u = \sin{\left(x \right)}$ .
  2. Apply the power rule: $u^{4}$ goes to $4 u^{3}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
    1. The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    The result of the chain rule is:
    
    $4 \sin^{3}{\left(x \right)} \cos{\left(x \right)}$
  The result of the chain rule is:
  
  $\frac{4 \cos{\left(x \right)}}{\sin{\left(x \right)}}$
So, the result is: $\frac{4 \cos{\left(x \right)}}{\log{\left(5 \right)} \sin{\left(x \right)}}$
Now simplify:

$\frac{4}{\log{\left(5 \right)} \tan{\left(x \right)}}$

The answer is:

$\frac{4}{\log{\left(5 \right)} \tan{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

$$\frac{4 \cos{\left(x \right)}}{\log{\left(5 \right)} \sin{\left(x \right)}}$$

Simplify

The second derivative [src]

   /       2   \
   |    cos (x)|
-4*|1 + -------|
   |       2   |
   \    sin (x)/
----------------
     log(5)

$$- \frac{4 \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right)}{\log{\left(5 \right)}}$$

Simplify

The third derivative [src]

  /       2   \       
  |    cos (x)|       
8*|1 + -------|*cos(x)
  |       2   |       
  \    sin (x)/       
----------------------
    log(5)*sin(x)

$$\frac{8 \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\log{\left(5 \right)} \sin{\left(x \right)}}$$

Simplify

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Derivative of log((sin(x))^4)/log(5)

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The solution