Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of 2/e^x
Derivative of y=3x^4
Derivative of 7*x+4
Derivative of (1-2*x)*cos(x)+2*sin(x)+3
Identical expressions
- four /cos^(two)(two -4y)
minus 4 divide by co sinus of e of to the power of (2)(2 minus 4y)
minus four divide by co sinus of e of to the power of (two)(two minus 4y)
-4/cos(2)(2-4y)
-4/cos22-4y
-4/cos^22-4y
-4 divide by cos^(2)(2-4y)
Similar expressions
4/cos^(2)(2-4y)
-4/cos^(2)(2+4y)

The derivative of the function/ -4/cos^(2)(2-4y)

Derivative of -4/cos^(2)(2-4y)

The solution

You have entered [src]

     -4      
-------------
   2         
cos (2 - 4*y)

$$- \frac{4}{\cos^{2}{\left(2 - 4 y \right)}}$$

-4/cos(2 - 4*y)^2

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \cos^{2}{\left(2 - 4 y \right)}$ .
2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d y} \cos^{2}{\left(2 - 4 y \right)}$ :
  1. Let $u = \cos{\left(2 - 4 y \right)}$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d y} \cos{\left(2 - 4 y \right)}$ :
    1. Let $u = 2 - 4 y$ .
    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 $\frac{d}{d y} \left(2 - 4 y\right)$ :
      1. Differentiate $2 - 4 y$ term by term:
        
        The derivative of the constant $2$ is zero.
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $y$ goes to $1$
        
        So, the result is: $-4$
        
        The result is: $-4$
      The result of the chain rule is:
      
      $- 4 \sin{\left(4 y - 2 \right)}$
    The result of the chain rule is:
    
    $- 8 \sin{\left(4 y - 2 \right)} \cos{\left(2 - 4 y \right)}$
  The result of the chain rule is:
  
  $\frac{8 \sin{\left(4 y - 2 \right)}}{\cos^{3}{\left(2 - 4 y \right)}}$
So, the result is: $- \frac{32 \sin{\left(4 y - 2 \right)}}{\cos^{3}{\left(2 - 4 y \right)}}$
Now simplify:

$- \frac{32 \sin{\left(4 y - 2 \right)}}{\cos^{3}{\left(4 y - 2 \right)}}$

The answer is:

$- \frac{32 \sin{\left(4 y - 2 \right)}}{\cos^{3}{\left(4 y - 2 \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

-32*sin(-2 + 4*y)
-----------------
     3           
  cos (2 - 4*y)

$$- \frac{32 \sin{\left(4 y - 2 \right)}}{\cos^{3}{\left(2 - 4 y \right)}}$$

Simplify

The second derivative [src]

     /         2              \
     |    3*sin (2*(-1 + 2*y))|
-128*|1 + --------------------|
     |        2               |
     \     cos (2*(-1 + 2*y)) /
-------------------------------
          2                    
       cos (2*(-1 + 2*y))

$$- \frac{128 \left(\frac{3 \sin^{2}{\left(2 \left(2 y - 1\right) \right)}}{\cos^{2}{\left(2 \left(2 y - 1\right) \right)}} + 1\right)}{\cos^{2}{\left(2 \left(2 y - 1\right) \right)}}$$

Simplify

The third derivative [src]

      /         2              \                  
      |    3*sin (2*(-1 + 2*y))|                  
-2048*|2 + --------------------|*sin(2*(-1 + 2*y))
      |        2               |                  
      \     cos (2*(-1 + 2*y)) /                  
--------------------------------------------------
                   3                              
                cos (2*(-1 + 2*y))

$$- \frac{2048 \left(\frac{3 \sin^{2}{\left(2 \left(2 y - 1\right) \right)}}{\cos^{2}{\left(2 \left(2 y - 1\right) \right)}} + 2\right) \sin{\left(2 \left(2 y - 1\right) \right)}}{\cos^{3}{\left(2 \left(2 y - 1\right) \right)}}$$

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of -4/cos^(2)(2-4y)

The graph:

Piecewise:

The solution