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Identical expressions
(one / four)*tan(x)^(four)
(1 divide by 4) multiply by tangent of (x) to the power of (4)
(one divide by four) multiply by tangent of (x) to the power of (four)
(1/4)*tan(x)(4)
1/4*tanx4
(1/4)tan(x)^(4)
(1/4)tan(x)(4)
1/4tanx4
1/4tanx^4
(1 divide by 4)*tan(x)^(4)

The derivative of the function/ (1/4)*tan(x)^(4)

Derivative of (1/4)*tan(x)^(4)

The solution

You have entered [src]

   4   
tan (x)
-------
   4

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

tan(x)^4/4

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: $u^{4}$ goes to $4 u^{3}$
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:
  
  $\frac{4 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{3}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
So, the result is: $\frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{3}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
Now simplify:

$\frac{\tan^{3}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$

The answer is:

$\frac{\tan^{3}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   3    /         2   \
tan (x)*\4 + 4*tan (x)/
-----------------------
           4

$$\frac{\left(4 \tan^{2}{\left(x \right)} + 4\right) \tan^{3}{\left(x \right)}}{4}$$

Simplify

The second derivative [src]

   2    /       2   \ /         2   \
tan (x)*\1 + tan (x)/*\3 + 5*tan (x)/

$$\left(\tan^{2}{\left(x \right)} + 1\right) \left(5 \tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)}$$

Simplify

The third derivative [src]

                /                           2                           \       
  /       2   \ |     4        /       2   \          2    /       2   \|       
2*\1 + tan (x)/*\2*tan (x) + 3*\1 + tan (x)/  + 10*tan (x)*\1 + tan (x)//*tan(x)

$$2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 10 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 2 \tan^{4}{\left(x \right)}\right) \tan{\left(x \right)}$$

Simplify

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Derivative of (1/4)*tan(x)^(4)

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The solution