Derivative of y(x)=5sin2x-2ctg5x

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5*sin(2*x) - 2*cot(5*x)

$$5 \sin{\left(2 x \right)} - 2 \cot{\left(5 x \right)}$$

d                          
--(5*sin(2*x) - 2*cot(5*x))
dx

$$\frac{d}{d x} \left(5 \sin{\left(2 x \right)} - 2 \cot{\left(5 x \right)}\right)$$

Transform

Detail solution

Differentiate $5 \sin{\left(2 x \right)} - 2 \cot{\left(5 x \right)}$ term by term:
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = 2 x$ .
  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 $\frac{d}{d x} 2 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $2$
    The result of the chain rule is:
    
    $2 \cos{\left(2 x \right)}$
  So, the result is: $10 \cos{\left(2 x \right)}$
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. There are multiple ways to do this derivative.
      Method #1
      
      Rewrite the function to be differentiated:
      
      $\cot{\left(5 x \right)} = \frac{1}{\tan{\left(5 x \right)}}$
      
      Let $u = \tan{\left(5 x \right)}$ .
      
      Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(5 x \right)}$ :
      
      Rewrite the function to be differentiated:
      
      $\tan{\left(5 x \right)} = \frac{\sin{\left(5 x \right)}}{\cos{\left(5 x \right)}}$
      
      Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \sin{\left(5 x \right)}$ and $g{\left(x \right)} = \cos{\left(5 x \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      
      Let $u = 5 x$ .
      
      The derivative of sine is cosine:
      
      $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$ :
      
      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: $5$
      
      The result of the chain rule is:
      
      $5 \cos{\left(5 x \right)}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      Let $u = 5 x$ .
      
      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 $\frac{d}{d x} 5 x$ :
      
      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: $5$
      
      The result of the chain rule is:
      
      $- 5 \sin{\left(5 x \right)}$
      
      Now plug in to the quotient rule:
      
      $\frac{5 \sin^{2}{\left(5 x \right)} + 5 \cos^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}}$
      
      The result of the chain rule is:
      
      $- \frac{5 \sin^{2}{\left(5 x \right)} + 5 \cos^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)} \tan^{2}{\left(5 x \right)}}$
      Method #2
      
      Rewrite the function to be differentiated:
      
      $\cot{\left(5 x \right)} = \frac{\cos{\left(5 x \right)}}{\sin{\left(5 x \right)}}$
      
      Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \cos{\left(5 x \right)}$ and $g{\left(x \right)} = \sin{\left(5 x \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      
      Let $u = 5 x$ .
      
      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 $\frac{d}{d x} 5 x$ :
      
      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: $5$
      
      The result of the chain rule is:
      
      $- 5 \sin{\left(5 x \right)}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      Let $u = 5 x$ .
      
      The derivative of sine is cosine:
      
      $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$ :
      
      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: $5$
      
      The result of the chain rule is:
      
      $5 \cos{\left(5 x \right)}$
      
      Now plug in to the quotient rule:
      
      $\frac{- 5 \sin^{2}{\left(5 x \right)} - 5 \cos^{2}{\left(5 x \right)}}{\sin^{2}{\left(5 x \right)}}$
    So, the result is: $- \frac{2 \cdot \left(5 \sin^{2}{\left(5 x \right)} + 5 \cos^{2}{\left(5 x \right)}\right)}{\cos^{2}{\left(5 x \right)} \tan^{2}{\left(5 x \right)}}$
  So, the result is: $\frac{2 \cdot \left(5 \sin^{2}{\left(5 x \right)} + 5 \cos^{2}{\left(5 x \right)}\right)}{\cos^{2}{\left(5 x \right)} \tan^{2}{\left(5 x \right)}}$
The result is: $\frac{2 \cdot \left(5 \sin^{2}{\left(5 x \right)} + 5 \cos^{2}{\left(5 x \right)}\right)}{\cos^{2}{\left(5 x \right)} \tan^{2}{\left(5 x \right)}} + 10 \cos{\left(2 x \right)}$
Now simplify:

$10 \cos{\left(2 x \right)} + \frac{20}{1 - \cos{\left(10 x \right)}}$

The answer is:

$10 \cos{\left(2 x \right)} + \frac{20}{1 - \cos{\left(10 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

           2                   
10 + 10*cot (5*x) + 10*cos(2*x)

$$10 \cos{\left(2 x \right)} + 10 \cot^{2}{\left(5 x \right)} + 10$$

Simplify

The second derivative [src]

    /  /       2     \                    \
-20*\5*\1 + cot (5*x)/*cot(5*x) + sin(2*x)/

$$- 20 \cdot \left(5 \left(\cot^{2}{\left(5 x \right)} + 1\right) \cot{\left(5 x \right)} + \sin{\left(2 x \right)}\right)$$

Simplify

The third derivative [src]

   /                                2                               \
   |                 /       2     \          2      /       2     \|
20*\-2*cos(2*x) + 25*\1 + cot (5*x)/  + 50*cot (5*x)*\1 + cot (5*x)//

$$20 \cdot \left(25 \left(\cot^{2}{\left(5 x \right)} + 1\right)^{2} + 50 \left(\cot^{2}{\left(5 x \right)} + 1\right) \cot^{2}{\left(5 x \right)} - 2 \cos{\left(2 x \right)}\right)$$

Simplify

The graph

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Derivative of y(x)=5sin2x-2ctg5x

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The solution

Method #1

Method #2