Derivative of sec²(4x)+tan²(4x)

The solution

You have entered [src]

   2           2     
sec (4*x) + tan (4*x)

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

sec(4*x)^2 + tan(4*x)^2

Detail solution

Differentiate $\tan^{2}{\left(4 x \right)} + \sec^{2}{\left(4 x \right)}$ term by term:
1. Let $u = \sec{\left(4 x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sec{\left(4 x \right)}$ :
  1. Rewrite the function to be differentiated:
    
    $\sec{\left(4 x \right)} = \frac{1}{\cos{\left(4 x \right)}}$
  2. Let $u = \cos{\left(4 x \right)}$ .
  3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(4 x \right)}$ :
    1. Let $u = 4 x$ .
    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 $\frac{d}{d x} 4 x$ :
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $4$
      The result of the chain rule is:
      
      $- 4 \sin{\left(4 x \right)}$
    The result of the chain rule is:
    
    $\frac{4 \sin{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)}}$
  The result of the chain rule is:
  
  $\frac{8 \sin{\left(4 x \right)} \sec{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)}}$
4. Let $u = \tan{\left(4 x \right)}$ .
5. Apply the power rule: $u^{2}$ goes to $2 u$
6. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(4 x \right)}$ :
  1. Rewrite the function to be differentiated:
    
    $\tan{\left(4 x \right)} = \frac{\sin{\left(4 x \right)}}{\cos{\left(4 x \right)}}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin{\left(4 x \right)}$ and $g{\left(x \right)} = \cos{\left(4 x \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Let $u = 4 x$ .
    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 $\frac{d}{d x} 4 x$ :
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $4$
      The result of the chain rule is:
      
      $4 \cos{\left(4 x \right)}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = 4 x$ .
    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 $\frac{d}{d x} 4 x$ :
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $4$
      The result of the chain rule is:
      
      $- 4 \sin{\left(4 x \right)}$
    Now plug in to the quotient rule:
    
    $\frac{4 \sin^{2}{\left(4 x \right)} + 4 \cos^{2}{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)}}$
  The result of the chain rule is:
  
  $\frac{2 \left(4 \sin^{2}{\left(4 x \right)} + 4 \cos^{2}{\left(4 x \right)}\right) \tan{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)}}$
The result is: $\frac{2 \left(4 \sin^{2}{\left(4 x \right)} + 4 \cos^{2}{\left(4 x \right)}\right) \tan{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)}} + \frac{8 \sin{\left(4 x \right)} \sec{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)}}$
Now simplify:

$\frac{16 \tan{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)}}$

The answer is:

$\frac{16 \tan{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

/         2     \                 2              
\8 + 8*tan (4*x)/*tan(4*x) + 8*sec (4*x)*tan(4*x)

$$\left(8 \tan^{2}{\left(4 x \right)} + 8\right) \tan{\left(4 x \right)} + 8 \tan{\left(4 x \right)} \sec^{2}{\left(4 x \right)}$$

Simplify

The second derivative [src]

   /               2                                                                                  \
   |/       2     \       2      /       2     \        2         2             2      /       2     \|
32*\\1 + tan (4*x)/  + sec (4*x)*\1 + tan (4*x)/ + 2*sec (4*x)*tan (4*x) + 2*tan (4*x)*\1 + tan (4*x)//

$$32 \left(\left(\tan^{2}{\left(4 x \right)} + 1\right)^{2} + 2 \left(\tan^{2}{\left(4 x \right)} + 1\right) \tan^{2}{\left(4 x \right)} + \left(\tan^{2}{\left(4 x \right)} + 1\right) \sec^{2}{\left(4 x \right)} + 2 \tan^{2}{\left(4 x \right)} \sec^{2}{\left(4 x \right)}\right)$$

Simplify

The third derivative [src]

    /                 2                                                                                \         
    |  /       2     \       2         2           2      /       2     \        2      /       2     \|         
512*\2*\1 + tan (4*x)/  + sec (4*x)*tan (4*x) + tan (4*x)*\1 + tan (4*x)/ + 2*sec (4*x)*\1 + tan (4*x)//*tan(4*x)

$$512 \left(2 \left(\tan^{2}{\left(4 x \right)} + 1\right)^{2} + \left(\tan^{2}{\left(4 x \right)} + 1\right) \tan^{2}{\left(4 x \right)} + 2 \left(\tan^{2}{\left(4 x \right)} + 1\right) \sec^{2}{\left(4 x \right)} + \tan^{2}{\left(4 x \right)} \sec^{2}{\left(4 x \right)}\right) \tan{\left(4 x \right)}$$

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of sec²(4x)+tan²(4x)

The graph:

Piecewise:

The solution