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Identical expressions
y=tg^ five (3x^ four - thirteen)
y equally tg to the power of 5(3x to the power of 4 minus 13)
y equally tg to the power of five (3x to the power of four minus thirteen)
y=tg5(3x4-13)
y=tg53x4-13
y=tg⁵(3x⁴-13)
y=tg^53x^4-13
Similar expressions
y=tg^5(3x^4+13)

The derivative of the function/ y=tg^5(3x^4-13)

Derivative of y=tg^5(3x^4-13)

The solution

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   5/   4     \
tan \3*x  - 13/

$$\tan^{5}{\left(3 x^{4} - 13 \right)}$$

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Detail solution

Let $u = \tan{\left(3 x^{4} - 13 \right)}$ .
Apply the power rule: $u^{5}$ goes to $5 u^{4}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(3 x^{4} - 13 \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(3 x^{4} - 13 \right)} = \frac{\sin{\left(3 x^{4} - 13 \right)}}{\cos{\left(3 x^{4} - 13 \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(3 x^{4} - 13 \right)}$ and $g{\left(x \right)} = \cos{\left(3 x^{4} - 13 \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = 3 x^{4} - 13$ .
  2. The derivative of sine is cosine:
    
    $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(3 x^{4} - 13\right)$ :
    1. Differentiate $3 x^{4} - 13$ term by term:
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x^{4}$ goes to $4 x^{3}$
        
        So, the result is: $12 x^{3}$
      2. The derivative of the constant $-13$ is zero.
      The result is: $12 x^{3}$
    The result of the chain rule is:
    
    $12 x^{3} \cos{\left(3 x^{4} - 13 \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = 3 x^{4} - 13$ .
  2. The derivative of cosine is negative sine:
    
    $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(3 x^{4} - 13\right)$ :
    1. Differentiate $3 x^{4} - 13$ term by term:
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x^{4}$ goes to $4 x^{3}$
        
        So, the result is: $12 x^{3}$
      2. The derivative of the constant $-13$ is zero.
      The result is: $12 x^{3}$
    The result of the chain rule is:
    
    $- 12 x^{3} \sin{\left(3 x^{4} - 13 \right)}$
  Now plug in to the quotient rule:
  
  $\frac{12 x^{3} \sin^{2}{\left(3 x^{4} - 13 \right)} + 12 x^{3} \cos^{2}{\left(3 x^{4} - 13 \right)}}{\cos^{2}{\left(3 x^{4} - 13 \right)}}$
The result of the chain rule is:

$\frac{5 \left(12 x^{3} \sin^{2}{\left(3 x^{4} - 13 \right)} + 12 x^{3} \cos^{2}{\left(3 x^{4} - 13 \right)}\right) \tan^{4}{\left(3 x^{4} - 13 \right)}}{\cos^{2}{\left(3 x^{4} - 13 \right)}}$
Now simplify:

$\frac{60 x^{3} \tan^{4}{\left(3 x^{4} - 13 \right)}}{\cos^{2}{\left(3 x^{4} - 13 \right)}}$

The answer is:

$\frac{60 x^{3} \tan^{4}{\left(3 x^{4} - 13 \right)}}{\cos^{2}{\left(3 x^{4} - 13 \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    3    4/   4     \ /       2/   4     \\
60*x *tan \3*x  - 13/*\1 + tan \3*x  - 13//

$$60 x^{3} \left(\tan^{2}{\left(3 x^{4} - 13 \right)} + 1\right) \tan^{4}{\left(3 x^{4} - 13 \right)}$$

Simplify

The second derivative [src]

     2    3/         4\ /       2/         4\\ /   4    2/         4\       4 /       2/         4\\      /         4\\
180*x *tan \-13 + 3*x /*\1 + tan \-13 + 3*x //*\8*x *tan \-13 + 3*x / + 16*x *\1 + tan \-13 + 3*x // + tan\-13 + 3*x //

$$180 x^{2} \left(\tan^{2}{\left(3 x^{4} - 13 \right)} + 1\right) \left(16 x^{4} \left(\tan^{2}{\left(3 x^{4} - 13 \right)} + 1\right) + 8 x^{4} \tan^{2}{\left(3 x^{4} - 13 \right)} + \tan{\left(3 x^{4} - 13 \right)}\right) \tan^{3}{\left(3 x^{4} - 13 \right)}$$

Simplify

The third derivative [src]

                                              /                                                                                                  2                                                                                                \
         2/         4\ /       2/         4\\ |   2/         4\       4    3/         4\       8    4/         4\        8 /       2/         4\\        4 /       2/         4\\    /         4\        8    2/         4\ /       2/         4\\|
360*x*tan \-13 + 3*x /*\1 + tan \-13 + 3*x //*\tan \-13 + 3*x / + 36*x *tan \-13 + 3*x / + 96*x *tan \-13 + 3*x / + 288*x *\1 + tan \-13 + 3*x //  + 72*x *\1 + tan \-13 + 3*x //*tan\-13 + 3*x / + 624*x *tan \-13 + 3*x /*\1 + tan \-13 + 3*x ///

$$360 x \left(\tan^{2}{\left(3 x^{4} - 13 \right)} + 1\right) \left(288 x^{8} \left(\tan^{2}{\left(3 x^{4} - 13 \right)} + 1\right)^{2} + 624 x^{8} \left(\tan^{2}{\left(3 x^{4} - 13 \right)} + 1\right) \tan^{2}{\left(3 x^{4} - 13 \right)} + 96 x^{8} \tan^{4}{\left(3 x^{4} - 13 \right)} + 72 x^{4} \left(\tan^{2}{\left(3 x^{4} - 13 \right)} + 1\right) \tan{\left(3 x^{4} - 13 \right)} + 36 x^{4} \tan^{3}{\left(3 x^{4} - 13 \right)} + \tan^{2}{\left(3 x^{4} - 13 \right)}\right) \tan^{2}{\left(3 x^{4} - 13 \right)}$$

Simplify

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Derivative of y=tg^5(3x^4-13)

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