Derivative of y=sec²4x+tan²4x

The solution

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   24         24   
sec  (x) + tan  (x)

$$\tan^{24}{\left(x \right)} + \sec^{24}{\left(x \right)}$$

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Detail solution

Differentiate $\tan^{24}{\left(x \right)} + \sec^{24}{\left(x \right)}$ term by term:
1. Let $u = \sec{\left(x \right)}$ .
2. Apply the power rule: $u^{24}$ goes to $24 u^{23}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sec{\left(x \right)}$ :
  1. Rewrite the function to be differentiated:
    
    $\sec{\left(x \right)} = \frac{1}{\cos{\left(x \right)}}$
  2. Let $u = \cos{\left(x \right)}$ .
  3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    The result of the chain rule is:
    
    $\frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  The result of the chain rule is:
  
  $\frac{24 \sin{\left(x \right)} \sec^{23}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
4. Let $u = \tan{\left(x \right)}$ .
5. Apply the power rule: $u^{24}$ goes to $24 u^{23}$
6. 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:
  
  $\frac{24 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{23}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
The result is: $\frac{24 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{23}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{24 \sin{\left(x \right)} \sec^{23}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
Now simplify:

$\frac{24 \left(\sin^{22}{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos^{25}{\left(x \right)}}$

The answer is:

$\frac{24 \left(\sin^{22}{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos^{25}{\left(x \right)}}$

The graph

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The first derivative [src]

   23    /           2   \         24          
tan  (x)*\24 + 24*tan (x)/ + 24*sec  (x)*tan(x)

$$\left(24 \tan^{2}{\left(x \right)} + 24\right) \tan^{23}{\left(x \right)} + 24 \tan{\left(x \right)} \sec^{24}{\left(x \right)}$$

Simplify

The second derivative [src]

   /                                                                    2                               \
   |   24    /       2   \        24    /       2   \      /       2   \     22            24       2   |
24*\sec  (x)*\1 + tan (x)/ + 2*tan  (x)*\1 + tan (x)/ + 23*\1 + tan (x)/ *tan  (x) + 24*sec  (x)*tan (x)/

$$24 \left(23 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan^{22}{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{24}{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \sec^{24}{\left(x \right)} + 24 \tan^{2}{\left(x \right)} \sec^{24}{\left(x \right)}\right)$$

Simplify

The third derivative [src]

   /                                                                       2                             3                                \       
   |     24    /       2   \         24    /       2   \      /       2   \     22          /       2   \     20             24       2   |       
48*\2*tan  (x)*\1 + tan (x)/ + 37*sec  (x)*\1 + tan (x)/ + 70*\1 + tan (x)/ *tan  (x) + 253*\1 + tan (x)/ *tan  (x) + 288*sec  (x)*tan (x)/*tan(x)

$$48 \left(253 \left(\tan^{2}{\left(x \right)} + 1\right)^{3} \tan^{20}{\left(x \right)} + 70 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan^{22}{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{24}{\left(x \right)} + 37 \left(\tan^{2}{\left(x \right)} + 1\right) \sec^{24}{\left(x \right)} + 288 \tan^{2}{\left(x \right)} \sec^{24}{\left(x \right)}\right) \tan{\left(x \right)}$$

Simplify

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Derivative of y=sec²4x+tan²4x

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