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Derivative of:
Derivative of f(x)=4x^6+7x^5+1
Derivative of cot(cos(2))+((sin(6*x)^(2))/cos(12*x))*(1/6)
Derivative of (2*x+3)^5
Derivative of x^-tan(x)
Identical expressions
cot(cos(two))+((sin(six *x)^(two))/cos(twelve *x))*(one / six)
cotangent of ( co sinus of e of (2)) plus (( sinus of (6 multiply by x) to the power of (2)) divide by co sinus of e of (12 multiply by x)) multiply by (1 divide by 6)
cotangent of ( co sinus of e of (two)) plus (( sinus of (six multiply by x) to the power of (two)) divide by co sinus of e of (twelve multiply by x)) multiply by (one divide by six)
cot(cos(2))+((sin(6*x)(2))/cos(12*x))*(1/6)
cotcos2+sin6*x2/cos12*x*1/6
cot(cos(2))+((sin(6x)^(2))/cos(12x))(1/6)
cot(cos(2))+((sin(6x)(2))/cos(12x))(1/6)
cotcos2+sin6x2/cos12x1/6
cotcos2+sin6x^2/cos12x1/6
cot(cos(2))+((sin(6*x)^(2)) divide by cos(12*x))*(1 divide by 6)
Similar expressions
cot(cos(2))-((sin(6*x)^(2))/cos(12*x))*(1/6)

The derivative of the function/ cot(cos(2))+((sin(6*x)^(2))/cos(12*x))*(1/6)

Derivative of cot(cos(2))+((sin(6x)^(2))/cos(12x))*(1/6)

The solution

You have entered [src]

              /   2     \
              |sin (6*x)|
              |---------|
              \cos(12*x)/
cot(cos(2)) + -----------
                   6

$$\frac{\sin^{2}{\left(6 x \right)} \frac{1}{\cos{\left(12 x \right)}}}{6} + \cot{\left(\cos{\left(2 \right)} \right)}$$

cot(cos(2)) + (sin(6*x)^2/cos(12*x))/6

Detail solution

Differentiate $\frac{\sin^{2}{\left(6 x \right)} \frac{1}{\cos{\left(12 x \right)}}}{6} + \cot{\left(\cos{\left(2 \right)} \right)}$ term by term:
1. There are multiple ways to do this derivative.
  Method #1
  1. Rewrite the function to be differentiated:
    
    $\cot{\left(\cos{\left(2 \right)} \right)} = \frac{1}{\tan{\left(\cos{\left(2 \right)} \right)}}$
  2. The derivative of the constant $\frac{1}{\tan{\left(\cos{\left(2 \right)} \right)}}$ is zero.
  Method #2
  1. Rewrite the function to be differentiated:
    
    $\cot{\left(\cos{\left(2 \right)} \right)} = \frac{\cos{\left(\cos{\left(2 \right)} \right)}}{\sin{\left(\cos{\left(2 \right)} \right)}}$
  2. The derivative of the constant $\frac{\cos{\left(\cos{\left(2 \right)} \right)}}{\sin{\left(\cos{\left(2 \right)} \right)}}$ is zero.
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin^{2}{\left(6 x \right)}$ and $g{\left(x \right)} = \cos{\left(12 x \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Let $u = \sin{\left(6 x \right)}$ .
    2. Apply the power rule: $u^{2}$ goes to $2 u$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(6 x \right)}$ :
      1. Let $u = 6 x$ .
      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 $\frac{d}{d x} 6 x$ :
        
        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: $6$
        
        The result of the chain rule is:
        
        $6 \cos{\left(6 x \right)}$
      The result of the chain rule is:
      
      $12 \sin{\left(6 x \right)} \cos{\left(6 x \right)}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = 12 x$ .
    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 $\frac{d}{d x} 12 x$ :
      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: $12$
      The result of the chain rule is:
      
      $- 12 \sin{\left(12 x \right)}$
    Now plug in to the quotient rule:
    
    $\frac{12 \sin^{2}{\left(6 x \right)} \sin{\left(12 x \right)} + 12 \sin{\left(6 x \right)} \cos{\left(6 x \right)} \cos{\left(12 x \right)}}{\cos^{2}{\left(12 x \right)}}$
  So, the result is: $\frac{12 \sin^{2}{\left(6 x \right)} \sin{\left(12 x \right)} + 12 \sin{\left(6 x \right)} \cos{\left(6 x \right)} \cos{\left(12 x \right)}}{6 \cos^{2}{\left(12 x \right)}}$
The result is: $\frac{12 \sin^{2}{\left(6 x \right)} \sin{\left(12 x \right)} + 12 \sin{\left(6 x \right)} \cos{\left(6 x \right)} \cos{\left(12 x \right)}}{6 \cos^{2}{\left(12 x \right)}}$
Now simplify:

$\frac{2 \sin{\left(6 x \right)} \cos{\left(6 x \right)}}{\cos^{2}{\left(12 x \right)}}$

The answer is:

$\frac{2 \sin{\left(6 x \right)} \cos{\left(6 x \right)}}{\cos^{2}{\left(12 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                           2               
2*cos(6*x)*sin(6*x)   2*sin (6*x)*sin(12*x)
------------------- + ---------------------
     cos(12*x)                 2           
                            cos (12*x)

$$\frac{2 \sin^{2}{\left(6 x \right)} \sin{\left(12 x \right)}}{\cos^{2}{\left(12 x \right)}} + \frac{2 \sin{\left(6 x \right)} \cos{\left(6 x \right)}}{\cos{\left(12 x \right)}}$$

Simplify

The second derivative [src]

   /                             2         2                                      \
   |   2           2        4*sin (6*x)*sin (12*x)   4*cos(6*x)*sin(6*x)*sin(12*x)|
12*|cos (6*x) + sin (6*x) + ---------------------- + -----------------------------|
   |                                 2                         cos(12*x)          |
   \                              cos (12*x)                                      /
-----------------------------------------------------------------------------------
                                     cos(12*x)

$$\frac{12 \left(\frac{4 \sin^{2}{\left(6 x \right)} \sin^{2}{\left(12 x \right)}}{\cos^{2}{\left(12 x \right)}} + \sin^{2}{\left(6 x \right)} + \frac{4 \sin{\left(6 x \right)} \sin{\left(12 x \right)} \cos{\left(6 x \right)}}{\cos{\left(12 x \right)}} + \cos^{2}{\left(6 x \right)}\right)}{\cos{\left(12 x \right)}}$$

Simplify

The third derivative [src]

    /                           2                       2                        2         3               2                        \
    |                      3*cos (6*x)*sin(12*x)   7*sin (6*x)*sin(12*x)   12*sin (6*x)*sin (12*x)   12*sin (12*x)*cos(6*x)*sin(6*x)|
144*|4*cos(6*x)*sin(6*x) + --------------------- + --------------------- + ----------------------- + -------------------------------|
    |                            cos(12*x)               cos(12*x)                   3                             2                |
    \                                                                             cos (12*x)                    cos (12*x)          /
-------------------------------------------------------------------------------------------------------------------------------------
                                                              cos(12*x)

$$\frac{144 \left(\frac{12 \sin^{2}{\left(6 x \right)} \sin^{3}{\left(12 x \right)}}{\cos^{3}{\left(12 x \right)}} + \frac{7 \sin^{2}{\left(6 x \right)} \sin{\left(12 x \right)}}{\cos{\left(12 x \right)}} + \frac{12 \sin{\left(6 x \right)} \sin^{2}{\left(12 x \right)} \cos{\left(6 x \right)}}{\cos^{2}{\left(12 x \right)}} + 4 \sin{\left(6 x \right)} \cos{\left(6 x \right)} + \frac{3 \sin{\left(12 x \right)} \cos^{2}{\left(6 x \right)}}{\cos{\left(12 x \right)}}\right)}{\cos{\left(12 x \right)}}$$

Simplify

The graph

Derivative of cot(cos(2))+((sin(6*x)^(2))/cos(12*x))*(1/6)

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How to use it?

Identical expressions

Similar expressions

Derivative of cot(cos(2))+((sin(6x)^(2))/cos(12x))*(1/6)

The graph:

Piecewise:

The solution

Method #1

Method #2

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of cot(cos(2))+((sin(6*x)^(2))/cos(12*x))*(1/6)

The graph:

Piecewise:

The solution

Method #1

Method #2

Derivative of cot(cos(2))+((sin(6x)^(2))/cos(12x))*(1/6)