Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of x/(x+1)
Derivative of x/log(x)
Derivative of (x+1)/(x-1)
Derivative of (x^2-4)/x
Identical expressions
ctg√ five ^ three - one / eight *cos4x^ two /sin8x
ctg√5 cubed minus 1 divide by 8 multiply by co sinus of e of 4x squared divide by sinus of 8x
ctg√ five to the power of three minus one divide by eight multiply by co sinus of e of 4x to the power of two divide by sinus of 8x
ctg√53-1/8*cos4x2/sin8x
ctg√5³-1/8*cos4x²/sin8x
ctg√5 to the power of 3-1/8*cos4x to the power of 2/sin8x
ctg√5^3-1/8cos4x^2/sin8x
ctg√53-1/8cos4x2/sin8x
ctg√5^3-1 divide by 8*cos4x^2 divide by sin8x
Similar expressions
ctg√5^3+1/8*cos4x^2/sin8x

The derivative of the function/ ctg√5^3-1/8*cos4x^2/sin8x

Derivative of ctg√5^3-1/8*cos4x^2/sin8x

The solution

You have entered [src]

              /   2     \
              |cos (4*x)|
              |---------|
   3/  ___\   \    8    /
cot \\/ 5 / - -----------
                sin(8*x)

$$\cot^{3}{\left(\sqrt{5} \right)} - \frac{\frac{1}{8} \cos^{2}{\left(4 x \right)}}{\sin{\left(8 x \right)}}$$

cot(sqrt(5))^3 - cos(4*x)^2/8/sin(8*x)

Detail solution

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

$\frac{1}{64 \left(2 \sin^{2}{\left(x \right)} - 1\right)^{2} \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}$

The answer is:

$\frac{1}{64 \left(2 \sin^{2}{\left(x \right)} - 1\right)^{2} \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   2                                  
cos (4*x)*cos(8*x)   cos(4*x)*sin(4*x)
------------------ + -----------------
       2                  sin(8*x)    
    sin (8*x)

$$\frac{\sin{\left(4 x \right)} \cos{\left(4 x \right)}}{\sin{\left(8 x \right)}} + \frac{\cos^{2}{\left(4 x \right)} \cos{\left(8 x \right)}}{\sin^{2}{\left(8 x \right)}}$$

Simplify

The second derivative [src]

   /                             2         2                                    \
   |   2           2        4*cos (4*x)*cos (8*x)   4*cos(4*x)*cos(8*x)*sin(4*x)|
-4*|cos (4*x) + sin (4*x) + --------------------- + ----------------------------|
   |                                 2                        sin(8*x)          |
   \                              sin (8*x)                                     /
---------------------------------------------------------------------------------
                                     sin(8*x)

$$- \frac{4 \left(\sin^{2}{\left(4 x \right)} + \frac{4 \sin{\left(4 x \right)} \cos{\left(4 x \right)} \cos{\left(8 x \right)}}{\sin{\left(8 x \right)}} + \cos^{2}{\left(4 x \right)} + \frac{4 \cos^{2}{\left(4 x \right)} \cos^{2}{\left(8 x \right)}}{\sin^{2}{\left(8 x \right)}}\right)}{\sin{\left(8 x \right)}}$$

Simplify

The third derivative [src]

   /                           2                      2                       2         3              2                       \
   |                      3*sin (4*x)*cos(8*x)   7*cos (4*x)*cos(8*x)   12*cos (4*x)*cos (8*x)   12*cos (8*x)*cos(4*x)*sin(4*x)|
32*|4*cos(4*x)*sin(4*x) + -------------------- + -------------------- + ---------------------- + ------------------------------|
   |                            sin(8*x)               sin(8*x)                  3                            2                |
   \                                                                          sin (8*x)                    sin (8*x)           /
--------------------------------------------------------------------------------------------------------------------------------
                                                            sin(8*x)

$$\frac{32 \left(\frac{3 \sin^{2}{\left(4 x \right)} \cos{\left(8 x \right)}}{\sin{\left(8 x \right)}} + 4 \sin{\left(4 x \right)} \cos{\left(4 x \right)} + \frac{12 \sin{\left(4 x \right)} \cos{\left(4 x \right)} \cos^{2}{\left(8 x \right)}}{\sin^{2}{\left(8 x \right)}} + \frac{7 \cos^{2}{\left(4 x \right)} \cos{\left(8 x \right)}}{\sin{\left(8 x \right)}} + \frac{12 \cos^{2}{\left(4 x \right)} \cos^{3}{\left(8 x \right)}}{\sin^{3}{\left(8 x \right)}}\right)}{\sin{\left(8 x \right)}}$$

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of ctg√5^3-1/8*cos4x^2/sin8x

The graph:

Piecewise:

The solution