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1/cos^2(5x+1)

The derivative of the function/ 1/cos^2(5x-1)

Derivative of 1/cos^2(5x-1)

The solution

You have entered [src]

        1      
1*-------------
     2         
  cos (5*x - 1)

$$1 \cdot \frac{1}{\cos^{2}{\left(5 x - 1 \right)}}$$

d /        1      \
--|1*-------------|
dx|     2         |
  \  cos (5*x - 1)/

$$\frac{d}{d x} 1 \cdot \frac{1}{\cos^{2}{\left(5 x - 1 \right)}}$$

Transform

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = 1$ and $g{\left(x \right)} = \cos^{2}{\left(5 x - 1 \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of the constant $1$ is zero.
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \cos{\left(5 x - 1 \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(5 x - 1 \right)}$ :
  1. Let $u = 5 x - 1$ .
  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 $\frac{d}{d x} \left(5 x - 1\right)$ :
    1. Differentiate $5 x - 1$ 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: $x$ goes to $1$
        
        So, the result is: $5$
      2. The derivative of the constant $\left(-1\right) 1$ is zero.
      The result is: $5$
    The result of the chain rule is:
    
    $- 5 \sin{\left(5 x - 1 \right)}$
  The result of the chain rule is:
  
  $- 10 \sin{\left(5 x - 1 \right)} \cos{\left(5 x - 1 \right)}$
Now plug in to the quotient rule:

$\frac{10 \sin{\left(5 x - 1 \right)}}{\cos^{3}{\left(5 x - 1 \right)}}$
Now simplify:

$\frac{10 \sin{\left(5 x - 1 \right)}}{\cos^{3}{\left(5 x - 1 \right)}}$

The answer is:

$\frac{10 \sin{\left(5 x - 1 \right)}}{\cos^{3}{\left(5 x - 1 \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     10*sin(5*x - 1)      
--------------------------
                2         
cos(5*x - 1)*cos (5*x - 1)

$$\frac{10 \sin{\left(5 x - 1 \right)}}{\cos{\left(5 x - 1 \right)} \cos^{2}{\left(5 x - 1 \right)}}$$

Simplify

The second derivative [src]

   /         2          \
   |    3*sin (-1 + 5*x)|
50*|1 + ----------------|
   |        2           |
   \     cos (-1 + 5*x) /
-------------------------
         2               
      cos (-1 + 5*x)

$$\frac{50 \cdot \left(\frac{3 \sin^{2}{\left(5 x - 1 \right)}}{\cos^{2}{\left(5 x - 1 \right)}} + 1\right)}{\cos^{2}{\left(5 x - 1 \right)}}$$

Simplify

The third derivative [src]

     /         2          \              
     |    3*sin (-1 + 5*x)|              
1000*|2 + ----------------|*sin(-1 + 5*x)
     |        2           |              
     \     cos (-1 + 5*x) /              
-----------------------------------------
                 3                       
              cos (-1 + 5*x)

$$\frac{1000 \cdot \left(\frac{3 \sin^{2}{\left(5 x - 1 \right)}}{\cos^{2}{\left(5 x - 1 \right)}} + 2\right) \sin{\left(5 x - 1 \right)}}{\cos^{3}{\left(5 x - 1 \right)}}$$

Simplify

The graph

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Derivative of 1/cos^2(5x-1)

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Piecewise:

The solution