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Similar expressions
sinx/(5cosx-4)

The derivative of the function/ sinx/(5cosx+4)

Derivative of sinx/(5cosx+4)

The solution

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

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

sin(x)/(5*cos(x) + 4)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $5 \cos{\left(x \right)} + 4$ term by term:
  1. The derivative of the constant $4$ is zero.
  2. 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: $- 5 \sin{\left(x \right)}$
  The result is: $- 5 \sin{\left(x \right)}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

/       /       2             \               \       
|       | 10*sin (x)          |               |       
|     5*|------------ + cos(x)|               |       
|       \4 + 5*cos(x)         /    10*cos(x)  |       
|-1 + ------------------------- + ------------|*sin(x)
\            4 + 5*cos(x)         4 + 5*cos(x)/       
------------------------------------------------------
                     4 + 5*cos(x)

$$\frac{\left(\frac{5 \left(\cos{\left(x \right)} + \frac{10 \sin^{2}{\left(x \right)}}{5 \cos{\left(x \right)} + 4}\right)}{5 \cos{\left(x \right)} + 4} - 1 + \frac{10 \cos{\left(x \right)}}{5 \cos{\left(x \right)} + 4}\right) \sin{\left(x \right)}}{5 \cos{\left(x \right)} + 4}$$

Simplify

The third derivative [src]

                                   /                             2     \                                    
                              2    |      30*cos(x)       150*sin (x)  |      /       2             \       
                         5*sin (x)*|-1 + ------------ + ---------------|      | 10*sin (x)          |       
                 2                 |     4 + 5*cos(x)                 2|   15*|------------ + cos(x)|*cos(x)
           15*sin (x)              \                    (4 + 5*cos(x)) /      \4 + 5*cos(x)         /       
-cos(x) - ------------ + ----------------------------------------------- + ---------------------------------
          4 + 5*cos(x)                     4 + 5*cos(x)                               4 + 5*cos(x)          
------------------------------------------------------------------------------------------------------------
                                                4 + 5*cos(x)

$$\frac{\frac{15 \left(\cos{\left(x \right)} + \frac{10 \sin^{2}{\left(x \right)}}{5 \cos{\left(x \right)} + 4}\right) \cos{\left(x \right)}}{5 \cos{\left(x \right)} + 4} - \cos{\left(x \right)} + \frac{5 \left(-1 + \frac{30 \cos{\left(x \right)}}{5 \cos{\left(x \right)} + 4} + \frac{150 \sin^{2}{\left(x \right)}}{\left(5 \cos{\left(x \right)} + 4\right)^{2}}\right) \sin^{2}{\left(x \right)}}{5 \cos{\left(x \right)} + 4} - \frac{15 \sin^{2}{\left(x \right)}}{5 \cos{\left(x \right)} + 4}}{5 \cos{\left(x \right)} + 4}$$

Simplify

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Derivative of sinx/(5cosx+4)

The graph:

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The solution