Derivative of 10cos(x)/sin(x)

You have entered [src]

10*cos(x)
---------
  sin(x)

$$\frac{10 \cos{\left(x \right)}}{\sin{\left(x \right)}}$$

(10*cos(x))/sin(x)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 10 \cos{\left(x \right)}$ 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. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  So, the result is: $- 10 \sin{\left(x \right)}$
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:

$\frac{- 10 \sin^{2}{\left(x \right)} - 10 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
Now simplify:

$- \frac{10}{\sin^{2}{\left(x \right)}}$

The answer is:

$- \frac{10}{\sin^{2}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

            2   
      10*cos (x)
-10 - ----------
          2     
       sin (x)

$$-10 - \frac{10 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}$$

Simplify

The second derivative [src]

   /         2   \       
   |    2*cos (x)|       
10*|2 + ---------|*cos(x)
   |        2    |       
   \     sin (x) /       
-------------------------
          sin(x)

$$\frac{10 \left(2 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}}$$

Simplify

The third derivative [src]

    /                        /         2   \\
    |                   2    |    6*cos (x)||
    |                cos (x)*|5 + ---------||
    |         2              |        2    ||
    |    3*cos (x)           \     sin (x) /|
-10*|2 + --------- + -----------------------|
    |        2                  2           |
    \     sin (x)            sin (x)        /

$$- 10 \left(\frac{\left(5 + \frac{6 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 2 + \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right)$$

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of 10cos(x)/sin(x)

The graph:

Piecewise:

The solution