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Identical expressions
one / fifteen *tan(x)^(fifteen)
1 divide by 15 multiply by tangent of (x) to the power of (15)
one divide by fifteen multiply by tangent of (x) to the power of (fifteen)
1/15*tan(x)(15)
1/15*tanx15
1/15tan(x)^(15)
1/15tan(x)(15)
1/15tanx15
1/15tanx^15
1 divide by 15*tan(x)^(15)

The derivative of the function/ 1/15*tan(x)^(15)

Derivative of 1/15*tan(x)^(15)

The solution

You have entered [src]

   15   
tan  (x)
--------
   15

$$\frac{\tan^{15}{\left(x \right)}}{15}$$

  /   15   \
d |tan  (x)|
--|--------|
dx\   15   /

$$\frac{d}{d x} \frac{\tan^{15}{\left(x \right)}}{15}$$

Transform

Detail solution

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. Apply the power rule: $u^{15}$ goes to $15 u^{14}$
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 $\frac{d}{d x} f{\left(x \right)}$ :
    1. The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. 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:
  
  $\frac{15 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{14}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
So, the result is: $\frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{14}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
Now simplify:

$\frac{\tan^{14}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$

The answer is:

$\frac{\tan^{14}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   14    /           2   \
tan  (x)*\15 + 15*tan (x)/
--------------------------
            15

$$\frac{\left(15 \tan^{2}{\left(x \right)} + 15\right) \tan^{14}{\left(x \right)}}{15}$$

Simplify

The second derivative [src]

     13    /       2   \ /         2   \
2*tan  (x)*\1 + tan (x)/*\7 + 8*tan (x)/

$$2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(8 \tan^{2}{\left(x \right)} + 7\right) \tan^{13}{\left(x \right)}$$

Simplify

The third derivative [src]

                         /                            2                           \
     12    /       2   \ |     4         /       2   \          2    /       2   \|
2*tan  (x)*\1 + tan (x)/*\2*tan (x) + 91*\1 + tan (x)/  + 43*tan (x)*\1 + tan (x)//

$$2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(2 \tan^{4}{\left(x \right)} + 43 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 91 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}\right) \tan^{12}{\left(x \right)}$$

Simplify

The graph

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Derivative of 1/15*tan(x)^(15)

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The solution