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Integral of d{x}:
(sinx)/(cos^3x)
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Similar expressions
(2(sin^2x+cos^2x)sinx)/cos^3x

The derivative of the function/ (sinx)/(cos^3x)

Derivative of (sinx)/(cos^3x)

The solution

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 sin(x)
-------
   3   
cos (x)

$$\frac{\sin{\left(x \right)}}{\cos^{3}{\left(x \right)}}$$

sin(x)/cos(x)^3

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 $g{\left(x \right)} = \cos^{3}{\left(x \right)}$ .

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. Let $u = \cos{\left(x \right)}$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  The result of the chain rule is:
  
  $- 3 \sin{\left(x \right)} \cos^{2}{\left(x \right)}$
Now plug in to the quotient rule:

$\frac{3 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} + \cos^{4}{\left(x \right)}}{\cos^{6}{\left(x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

/          2   \       
|    12*sin (x)|       
|8 + ----------|*sin(x)
|        2     |       
\     cos (x)  /       
-----------------------
           3           
        cos (x)

$$\frac{\left(\frac{12 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 8\right) \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}}$$

Simplify

The third derivative [src]

                           /           2   \
                      2    |     20*sin (x)|
                 3*sin (x)*|11 + ----------|
          2                |         2     |
    27*sin (x)             \      cos (x)  /
8 + ---------- + ---------------------------
        2                     2             
     cos (x)               cos (x)          
--------------------------------------------
                     2                      
                  cos (x)

$$\frac{\frac{3 \left(\frac{20 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 11\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{27 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 8}{\cos^{2}{\left(x \right)}}$$

Simplify

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Derivative of (sinx)/(cos^3x)

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The solution