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1/cos^2(5x-1)

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

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

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)/
$$\frac{d}{d x} 1 \cdot \frac{1}{\cos^{2}{\left(5 x - 1 \right)}}$$
Detail solution
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. The derivative of the constant is zero.

    To find :

    1. Let .

    2. Apply the power rule: goes to

    3. Then, apply the chain rule. Multiply by :

      1. Let .

      2. The derivative of cosine is negative sine:

      3. Then, apply the chain rule. Multiply by :

        1. Differentiate term by term:

          1. The derivative of a constant times a function is the constant times the derivative of the function.

            1. Apply the power rule: goes to

            So, the result is:

          2. The derivative of the constant is zero.

          The result is:

        The result of the chain rule is:

      The result of the chain rule is:

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
     10*sin(5*x - 1)      
--------------------------
                2         
cos(5*x - 1)*cos (5*x - 1)
$$\frac{10 \sin{\left(5 x - 1 \right)}}{\cos{\left(5 x - 1 \right)} \cos^{2}{\left(5 x - 1 \right)}}$$
The second derivative [src]
   /         2          \
   |    3*sin (-1 + 5*x)|
50*|1 + ----------------|
   |        2           |
   \     cos (-1 + 5*x) /
-------------------------
         2               
      cos (-1 + 5*x)     
$$\frac{50 \cdot \left(\frac{3 \sin^{2}{\left(5 x - 1 \right)}}{\cos^{2}{\left(5 x - 1 \right)}} + 1\right)}{\cos^{2}{\left(5 x - 1 \right)}}$$
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)             
$$\frac{1000 \cdot \left(\frac{3 \sin^{2}{\left(5 x - 1 \right)}}{\cos^{2}{\left(5 x - 1 \right)}} + 2\right) \sin{\left(5 x - 1 \right)}}{\cos^{3}{\left(5 x - 1 \right)}}$$
The graph
Derivative of 1/cos^2(5x-1)