Derivative of log16(log5(tanx))

The solution

You have entered [src]

   /log(tan(x))\
log|-----------|
   \   log(5)  /
----------------
    log(16)

$$\frac{\log{\left(\frac{\log{\left(\tan{\left(x \right)} \right)}}{\log{\left(5 \right)}} \right)}}{\log{\left(16 \right)}}$$

  /   /log(tan(x))\\
  |log|-----------||
d |   \   log(5)  /|
--|----------------|
dx\    log(16)     /

$$\frac{d}{d x} \frac{\log{\left(\frac{\log{\left(\tan{\left(x \right)} \right)}}{\log{\left(5 \right)}} \right)}}{\log{\left(16 \right)}}$$

Transform

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \frac{\log{\left(\tan{\left(x \right)} \right)}}{\log{\left(5 \right)}}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{\log{\left(\tan{\left(x \right)} \right)}}{\log{\left(5 \right)}}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = \tan{\left(x \right)}$ .
    2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(x \right)}$ :
      1. Rewrite the function to be differentiated:
        
        $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
      2. Apply the quotient rule, which is:
        
        $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
        
        $f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = \cos{\left(x \right)}$ .
        
        To find $\frac{d}{d x} f{\left(x \right)}$ :
        
        The derivative of sine is cosine:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
        
        To find $\frac{d}{d x} g{\left(x \right)}$ :
        
        The derivative of cosine is negative sine:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
        
        Now plug in to the quotient rule:
        
        $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
      The result of the chain rule is:
      
      $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan{\left(x \right)}}$
    So, the result is: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\log{\left(5 \right)} \cos^{2}{\left(x \right)} \tan{\left(x \right)}}$
  The result of the chain rule is:
  
  $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\log{\left(\tan{\left(x \right)} \right)} \cos^{2}{\left(x \right)} \tan{\left(x \right)}}$
So, the result is: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\log{\left(16 \right)} \log{\left(\tan{\left(x \right)} \right)} \cos^{2}{\left(x \right)} \tan{\left(x \right)}}$
Now simplify:

$\frac{2}{\log{\left(16 \right)} \log{\left(\tan{\left(x \right)} \right)} \sin{\left(2 x \right)}}$

The answer is:

$\frac{2}{\log{\left(16 \right)} \log{\left(\tan{\left(x \right)} \right)} \sin{\left(2 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

              2           
       1 + tan (x)        
--------------------------
log(16)*log(tan(x))*tan(x)

$$\frac{\tan^{2}{\left(x \right)} + 1}{\log{\left(16 \right)} \log{\left(\tan{\left(x \right)} \right)} \tan{\left(x \right)}}$$

Simplify

The second derivative [src]

               /            2                 2       \ 
 /       2   \ |     1 + tan (x)       1 + tan (x)    | 
-\1 + tan (x)/*|-2 + ----------- + -------------------| 
               |          2                       2   | 
               \       tan (x)     log(tan(x))*tan (x)/ 
--------------------------------------------------------
                  log(16)*log(tan(x))

$$- \frac{\left(\tan^{2}{\left(x \right)} + 1\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)}} + \frac{\tan^{2}{\left(x \right)} + 1}{\log{\left(\tan{\left(x \right)} \right)} \tan^{2}{\left(x \right)}} - 2\right)}{\log{\left(16 \right)} \log{\left(\tan{\left(x \right)} \right)}}$$

Simplify

The third derivative [src]

              /                                            2                                         2                      2 \
              |             /       2   \     /       2   \       /       2   \         /       2   \          /       2   \  |
/       2   \ |           4*\1 + tan (x)/   2*\1 + tan (x)/     6*\1 + tan (x)/       2*\1 + tan (x)/        3*\1 + tan (x)/  |
\1 + tan (x)/*|4*tan(x) - --------------- + ---------------- - ------------------ + -------------------- + -------------------|
              |                tan(x)              3           log(tan(x))*tan(x)      2            3                     3   |
              \                                 tan (x)                             log (tan(x))*tan (x)   log(tan(x))*tan (x)/
-------------------------------------------------------------------------------------------------------------------------------
                                                      log(16)*log(tan(x))

$$\frac{\left(\tan^{2}{\left(x \right)} + 1\right) \left(\frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{3}{\left(x \right)}} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\log{\left(\tan{\left(x \right)} \right)} \tan^{3}{\left(x \right)}} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\log{\left(\tan{\left(x \right)} \right)}^{2} \tan^{3}{\left(x \right)}} - \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} - \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right)}{\log{\left(\tan{\left(x \right)} \right)} \tan{\left(x \right)}} + 4 \tan{\left(x \right)}\right)}{\log{\left(16 \right)} \log{\left(\tan{\left(x \right)} \right)}}$$

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Derivative of log16(log5(tanx))

The graph:

Piecewise:

The solution