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4*x/sinx

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

Derivative of 4*x/sin(x)

The solution

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 4*x  
------
sin(x)

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

(4*x)/sin(x)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
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: $4$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  4      4*x*cos(x)
------ - ----------
sin(x)       2     
          sin (x)

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

Simplify

The second derivative [src]

  /  /         2   \           \
  |  |    2*cos (x)|   2*cos(x)|
4*|x*|1 + ---------| - --------|
  |  |        2    |    sin(x) |
  \  \     sin (x) /           /
--------------------------------
             sin(x)

$$\frac{4 \left(x \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right)}{\sin{\left(x \right)}}$$

Simplify

The third derivative [src]

  /                  /         2   \       \
  |                  |    6*cos (x)|       |
  |                x*|5 + ---------|*cos(x)|
  |         2        |        2    |       |
  |    6*cos (x)     \     sin (x) /       |
4*|3 + --------- - ------------------------|
  |        2                sin(x)         |
  \     sin (x)                            /
--------------------------------------------
                   sin(x)

$$\frac{4 \left(- \frac{x \left(5 + \frac{6 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} + 3 + \frac{6 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right)}{\sin{\left(x \right)}}$$

Simplify

The graph

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Derivative of 4*x/sin(x)

The graph:

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The solution