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Derivative of:
Derivative of x^12
Derivative of -e^x
Derivative of e^(2-x)
Derivative of x^2-4*x
Limit of the function:
tan(5*x)/sin(4*x)
Identical expressions
tan(five *x)/sin(four *x)
tangent of (5 multiply by x) divide by sinus of (4 multiply by x)
tangent of (five multiply by x) divide by sinus of (four multiply by x)
tan(5x)/sin(4x)
tan5x/sin4x
tan(5*x) divide by sin(4*x)

The derivative of the function/ tan(5*x)/sin(4*x)

Derivative of tan(5x)/sin(4x)

The solution

You have entered [src]

tan(5*x)
--------
sin(4*x)

$$\frac{\tan{\left(5 x \right)}}{\sin{\left(4 x \right)}}$$

tan(5*x)/sin(4*x)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \tan{\left(5 x \right)}$ and $g{\left(x \right)} = \sin{\left(4 x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(5 x \right)} = \frac{\sin{\left(5 x \right)}}{\cos{\left(5 x \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(5 x \right)}$ and $g{\left(x \right)} = \cos{\left(5 x \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = 5 x$ .
  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 $\frac{d}{d x} 5 x$ :
    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:
    
    $5 \cos{\left(5 x \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  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:
    
    $- 5 \sin{\left(5 x \right)}$
  Now plug in to the quotient rule:
  
  $\frac{5 \sin^{2}{\left(5 x \right)} + 5 \cos^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 4 x$ .
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 $\frac{d}{d x} 4 x$ :
  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: $4$
  The result of the chain rule is:
  
  $4 \cos{\left(4 x \right)}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         2                           
5 + 5*tan (5*x)   4*cos(4*x)*tan(5*x)
--------------- - -------------------
    sin(4*x)              2          
                       sin (4*x)

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

Simplify

The second derivative [src]

  /  /         2     \                                             /       2     \         \
  |  |    2*cos (4*x)|               /       2     \            20*\1 + tan (5*x)/*cos(4*x)|
2*|8*|1 + -----------|*tan(5*x) + 25*\1 + tan (5*x)/*tan(5*x) - ---------------------------|
  |  |        2      |                                                    sin(4*x)         |
  \  \     sin (4*x) /                                                                     /
--------------------------------------------------------------------------------------------
                                          sin(4*x)

$$\frac{2 \left(8 \left(1 + \frac{2 \cos^{2}{\left(4 x \right)}}{\sin^{2}{\left(4 x \right)}}\right) \tan{\left(5 x \right)} + 25 \left(\tan^{2}{\left(5 x \right)} + 1\right) \tan{\left(5 x \right)} - \frac{20 \left(\tan^{2}{\left(5 x \right)} + 1\right) \cos{\left(4 x \right)}}{\sin{\left(4 x \right)}}\right)}{\sin{\left(4 x \right)}}$$

Simplify

The third derivative [src]

  /                                                                                                                           /         2     \                  \
  |                                                                                                                           |    6*cos (4*x)|                  |
  |                                                                                                                        32*|5 + -----------|*cos(4*x)*tan(5*x)|
  |                    /         2     \                                               /       2     \                        |        2      |                  |
  |    /       2     \ |    2*cos (4*x)|       /       2     \ /         2     \   300*\1 + tan (5*x)/*cos(4*x)*tan(5*x)      \     sin (4*x) /                  |
2*|120*\1 + tan (5*x)/*|1 + -----------| + 125*\1 + tan (5*x)/*\1 + 3*tan (5*x)/ - ------------------------------------- - --------------------------------------|
  |                    |        2      |                                                          sin(4*x)                                sin(4*x)               |
  \                    \     sin (4*x) /                                                                                                                         /
------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                             sin(4*x)

$$\frac{2 \left(120 \left(1 + \frac{2 \cos^{2}{\left(4 x \right)}}{\sin^{2}{\left(4 x \right)}}\right) \left(\tan^{2}{\left(5 x \right)} + 1\right) - \frac{32 \left(5 + \frac{6 \cos^{2}{\left(4 x \right)}}{\sin^{2}{\left(4 x \right)}}\right) \cos{\left(4 x \right)} \tan{\left(5 x \right)}}{\sin{\left(4 x \right)}} + 125 \left(\tan^{2}{\left(5 x \right)} + 1\right) \left(3 \tan^{2}{\left(5 x \right)} + 1\right) - \frac{300 \left(\tan^{2}{\left(5 x \right)} + 1\right) \cos{\left(4 x \right)} \tan{\left(5 x \right)}}{\sin{\left(4 x \right)}}\right)}{\sin{\left(4 x \right)}}$$

Simplify

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Identical expressions

Derivative of tan(5x)/sin(4x)

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Derivative of tan(5*x)/sin(4*x)

The graph:

Piecewise:

The solution

Derivative of tan(5x)/sin(4x)