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The derivative of the function/ 12*cosx/(1-sinx)

Derivative of 12*cosx/(1-sinx)

The solution

You have entered [src]

12*cos(x) 
----------
1 - sin(x)

$$\frac{12 \cos{\left(x \right)}}{1 - \sin{\left(x \right)}}$$

(12*cos(x))/(1 - sin(x))

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 12 \cos{\left(x \right)}$ and $g{\left(x \right)} = 1 - \sin{\left(x \right)}$ .

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: $- 12 \sin{\left(x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $1 - \sin{\left(x \right)}$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    So, the result is: $- \cos{\left(x \right)}$
  The result is: $- \cos{\left(x \right)}$
Now plug in to the quotient rule:

$\frac{- 12 \left(1 - \sin{\left(x \right)}\right) \sin{\left(x \right)} + 12 \cos^{2}{\left(x \right)}}{\left(1 - \sin{\left(x \right)}\right)^{2}}$
Now simplify:

$- \frac{12}{\sin{\left(x \right)} - 1}$

The answer is:

$- \frac{12}{\sin{\left(x \right)} - 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                       2    
  12*sin(x)      12*cos (x) 
- ---------- + -------------
  1 - sin(x)               2
               (1 - sin(x))

$$- \frac{12 \sin{\left(x \right)}}{1 - \sin{\left(x \right)}} + \frac{12 \cos^{2}{\left(x \right)}}{\left(1 - \sin{\left(x \right)}\right)^{2}}$$

Simplify

The second derivative [src]

   /          2                           \       
   |     2*cos (x)                        |       
   |    ----------- + sin(x)              |       
   |    -1 + sin(x)              2*sin(x) |       
12*|1 - -------------------- - -----------|*cos(x)
   \        -1 + sin(x)        -1 + sin(x)/       
--------------------------------------------------
                   -1 + sin(x)

$$\frac{12 \left(1 - \frac{\sin{\left(x \right)} + \frac{2 \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - 1}}{\sin{\left(x \right)} - 1} - \frac{2 \sin{\left(x \right)}}{\sin{\left(x \right)} - 1}\right) \cos{\left(x \right)}}{\sin{\left(x \right)} - 1}$$

Simplify

The third derivative [src]

   /                                /                          2      \                                  \
   |                           2    |       6*sin(x)      6*cos (x)   |     /      2             \       |
   |                        cos (x)*|-1 + ----------- + --------------|     | 2*cos (x)          |       |
   |                2               |     -1 + sin(x)                2|   3*|----------- + sin(x)|*sin(x)|
   |           3*cos (x)            \                   (-1 + sin(x)) /     \-1 + sin(x)         /       |
12*|-sin(x) - ----------- + ------------------------------------------- + -------------------------------|
   \          -1 + sin(x)                   -1 + sin(x)                             -1 + sin(x)          /
----------------------------------------------------------------------------------------------------------
                                               -1 + sin(x)

$$\frac{12 \left(- \sin{\left(x \right)} + \frac{3 \left(\sin{\left(x \right)} + \frac{2 \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{\sin{\left(x \right)} - 1} + \frac{\left(-1 + \frac{6 \sin{\left(x \right)}}{\sin{\left(x \right)} - 1} + \frac{6 \cos^{2}{\left(x \right)}}{\left(\sin{\left(x \right)} - 1\right)^{2}}\right) \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - 1} - \frac{3 \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - 1}\right)}{\sin{\left(x \right)} - 1}$$

Simplify

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Derivative of 12*cosx/(1-sinx)

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The solution