Derivative of y=ctg3x/4cos5x

The solution

You have entered [src]

cot(3*x)         
--------*cos(5*x)
   4

$$\frac{\cot{\left(3 x \right)}}{4} \cos{\left(5 x \right)}$$

(cot(3*x)/4)*cos(5*x)

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \cos{\left(5 x \right)} \cot{\left(3 x \right)}$ and $g{\left(x \right)} = 4$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \cos{\left(5 x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = 5 x$ .
  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 $\frac{d}{d x} 5 x$ :
    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: $5$
    The result of the chain rule is:
    
    $- 5 \sin{\left(5 x \right)}$
  $g{\left(x \right)} = \cot{\left(3 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. There are multiple ways to do this derivative.
    Method #1
    1. Rewrite the function to be differentiated:
      
      $\cot{\left(3 x \right)} = \frac{1}{\tan{\left(3 x \right)}}$
    2. Let $u = \tan{\left(3 x \right)}$ .
    3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
    4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(3 x \right)}$ :
      
      Rewrite the function to be differentiated:
      
      $\tan{\left(3 x \right)} = \frac{\sin{\left(3 x \right)}}{\cos{\left(3 x \right)}}$
      
      Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \sin{\left(3 x \right)}$ and $g{\left(x \right)} = \cos{\left(3 x \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      
      Let $u = 3 x$ .
      
      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 $\frac{d}{d x} 3 x$ :
      
      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: $3$
      
      The result of the chain rule is:
      
      $3 \cos{\left(3 x \right)}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      Let $u = 3 x$ .
      
      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 $\frac{d}{d x} 3 x$ :
      
      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: $3$
      
      The result of the chain rule is:
      
      $- 3 \sin{\left(3 x \right)}$
      
      Now plug in to the quotient rule:
      
      $\frac{3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}}$
      
      The result of the chain rule is:
      
      $- \frac{3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)} \tan^{2}{\left(3 x \right)}}$
    Method #2
    1. Rewrite the function to be differentiated:
      
      $\cot{\left(3 x \right)} = \frac{\cos{\left(3 x \right)}}{\sin{\left(3 x \right)}}$
    2. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \cos{\left(3 x \right)}$ and $g{\left(x \right)} = \sin{\left(3 x \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      
      Let $u = 3 x$ .
      
      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 $\frac{d}{d x} 3 x$ :
      
      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: $3$
      
      The result of the chain rule is:
      
      $- 3 \sin{\left(3 x \right)}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      Let $u = 3 x$ .
      
      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 $\frac{d}{d x} 3 x$ :
      
      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: $3$
      
      The result of the chain rule is:
      
      $3 \cos{\left(3 x \right)}$
      
      Now plug in to the quotient rule:
      
      $\frac{- 3 \sin^{2}{\left(3 x \right)} - 3 \cos^{2}{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)}}$
  The result is: $- \frac{\left(3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}\right) \cos{\left(5 x \right)}}{\cos^{2}{\left(3 x \right)} \tan^{2}{\left(3 x \right)}} - 5 \sin{\left(5 x \right)} \cot{\left(3 x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of the constant $4$ is zero.
Now plug in to the quotient rule:

$- \frac{\left(3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}\right) \cos{\left(5 x \right)}}{4 \cos^{2}{\left(3 x \right)} \tan^{2}{\left(3 x \right)}} - \frac{5 \sin{\left(5 x \right)} \cot{\left(3 x \right)}}{4}$
Now simplify:

$\frac{- 5 \cos{\left(x \right)} - 12 \cos{\left(5 x \right)} + 5 \cos{\left(11 x \right)}}{16 \sin^{2}{\left(3 x \right)}}$

The answer is:

$\frac{- 5 \cos{\left(x \right)} - 12 \cos{\left(5 x \right)} + 5 \cos{\left(11 x \right)}}{16 \sin^{2}{\left(3 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

                           /       2     \               /       2     \                  
-25*cos(5*x)*cot(3*x) + 30*\1 + cot (3*x)/*sin(5*x) + 18*\1 + cot (3*x)/*cos(5*x)*cot(3*x)
------------------------------------------------------------------------------------------
                                            4

$$\frac{30 \left(\cot^{2}{\left(3 x \right)} + 1\right) \sin{\left(5 x \right)} + 18 \left(\cot^{2}{\left(3 x \right)} + 1\right) \cos{\left(5 x \right)} \cot{\left(3 x \right)} - 25 \cos{\left(5 x \right)} \cot{\left(3 x \right)}}{4}$$

Simplify

The third derivative [src]

                            /       2     \                /       2     \                        /       2     \ /         2     \         
125*cot(3*x)*sin(5*x) + 225*\1 + cot (3*x)/*cos(5*x) - 270*\1 + cot (3*x)/*cot(3*x)*sin(5*x) - 54*\1 + cot (3*x)/*\1 + 3*cot (3*x)/*cos(5*x)
--------------------------------------------------------------------------------------------------------------------------------------------
                                                                     4

$$\frac{- 54 \left(\cot^{2}{\left(3 x \right)} + 1\right) \left(3 \cot^{2}{\left(3 x \right)} + 1\right) \cos{\left(5 x \right)} - 270 \left(\cot^{2}{\left(3 x \right)} + 1\right) \sin{\left(5 x \right)} \cot{\left(3 x \right)} + 225 \left(\cot^{2}{\left(3 x \right)} + 1\right) \cos{\left(5 x \right)} + 125 \sin{\left(5 x \right)} \cot{\left(3 x \right)}}{4}$$

Simplify

Other calculators

How to use it?

Identical expressions

Derivative of y=ctg3x/4cos5x

The graph:

Piecewise:

The solution

Method #1

Method #2