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Limit of the function:
tan(4*x)/sin(3*x)
Identical expressions
tan(four *x)/sin(three *x)
tangent of (4 multiply by x) divide by sinus of (3 multiply by x)
tangent of (four multiply by x) divide by sinus of (three multiply by x)
tan(4x)/sin(3x)
tan4x/sin3x
tan(4*x) divide by sin(3*x)

The derivative of the function/ tan(4*x)/sin(3*x)

Derivative of tan(4x)/sin(3x)

The solution

You have entered [src]

tan(4*x)
--------
sin(3*x)

$$\frac{\tan{\left(4 x \right)}}{\sin{\left(3 x \right)}}$$

tan(4*x)/sin(3*x)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \tan{\left(4 x \right)}$ and $g{\left(x \right)} = \sin{\left(3 x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(4 x \right)} = \frac{\sin{\left(4 x \right)}}{\cos{\left(4 x \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(4 x \right)}$ and $g{\left(x \right)} = \cos{\left(4 x \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  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:
    
    $4 \cos{\left(4 x \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = 4 x$ .
  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 $\frac{d}{d x} 4 x$ :
    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:
    
    $- 4 \sin{\left(4 x \right)}$
  Now plug in to the quotient rule:
  
  $\frac{4 \sin^{2}{\left(4 x \right)} + 4 \cos^{2}{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 3 x$ .
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 $\frac{d}{d x} 3 x$ :
  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: $3$
  The result of the chain rule is:
  
  $3 \cos{\left(3 x \right)}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         2                           
4 + 4*tan (4*x)   3*cos(3*x)*tan(4*x)
--------------- - -------------------
    sin(3*x)              2          
                       sin (3*x)

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

Simplify

The second derivative [src]

  /         2     \                                             /       2     \         
  |    2*cos (3*x)|               /       2     \            24*\1 + tan (4*x)/*cos(3*x)
9*|1 + -----------|*tan(4*x) + 32*\1 + tan (4*x)/*tan(4*x) - ---------------------------
  |        2      |                                                    sin(3*x)         
  \     sin (3*x) /                                                                     
----------------------------------------------------------------------------------------
                                        sin(3*x)

$$\frac{9 \left(1 + \frac{2 \cos^{2}{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)}}\right) \tan{\left(4 x \right)} + 32 \left(\tan^{2}{\left(4 x \right)} + 1\right) \tan{\left(4 x \right)} - \frac{24 \left(\tan^{2}{\left(4 x \right)} + 1\right) \cos{\left(3 x \right)}}{\sin{\left(3 x \right)}}}{\sin{\left(3 x \right)}}$$

Simplify

The third derivative [src]

                                                                                                                           /         2     \                  
                                                                                                                           |    6*cos (3*x)|                  
                                                                                                                        27*|5 + -----------|*cos(3*x)*tan(4*x)
                    /         2     \                                               /       2     \                        |        2      |                  
    /       2     \ |    2*cos (3*x)|       /       2     \ /         2     \   288*\1 + tan (4*x)/*cos(3*x)*tan(4*x)      \     sin (3*x) /                  
108*\1 + tan (4*x)/*|1 + -----------| + 128*\1 + tan (4*x)/*\1 + 3*tan (4*x)/ - ------------------------------------- - --------------------------------------
                    |        2      |                                                          sin(3*x)                                sin(3*x)               
                    \     sin (3*x) /                                                                                                                         
--------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                           sin(3*x)

$$\frac{108 \left(1 + \frac{2 \cos^{2}{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)}}\right) \left(\tan^{2}{\left(4 x \right)} + 1\right) - \frac{27 \left(5 + \frac{6 \cos^{2}{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)}}\right) \cos{\left(3 x \right)} \tan{\left(4 x \right)}}{\sin{\left(3 x \right)}} + 128 \left(\tan^{2}{\left(4 x \right)} + 1\right) \left(3 \tan^{2}{\left(4 x \right)} + 1\right) - \frac{288 \left(\tan^{2}{\left(4 x \right)} + 1\right) \cos{\left(3 x \right)} \tan{\left(4 x \right)}}{\sin{\left(3 x \right)}}}{\sin{\left(3 x \right)}}$$

Simplify

The graph

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Identical expressions

Derivative of tan(4x)/sin(3x)

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Derivative of tan(4*x)/sin(3*x)

The graph:

Piecewise:

The solution

Derivative of tan(4x)/sin(3x)