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Derivative of:
Derivative of sin^2x/cosx
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sin^2x/cosx
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sin^2x/cosx
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sin^2x/cosx
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The derivative of the function/ sin^2x/cosx

Derivative of sin^2x/cosx

The solution

You have entered [src]

   2   
sin (x)
-------
 cos(x)

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

  /   2   \
d |sin (x)|
--|-------|
dx\ cos(x)/

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

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sin^{2}{\left(x \right)}$ and $g{\left(x \right)} = \cos{\left(x \right)}$ .

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

/                                                  /         2   \\       
|                                             2    |    6*sin (x)||       
|                                          sin (x)*|5 + ---------||       
|       /   2         2   \         2              |        2    ||       
|     6*\sin (x) - cos (x)/   12*sin (x)           \     cos (x) /|       
|-2 - --------------------- + ---------- + -----------------------|*sin(x)
|               2                 2                   2           |       
\            cos (x)           cos (x)             cos (x)        /

$$\left(\frac{\left(\frac{6 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 5\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{12 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - 2 - \frac{6 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}}\right) \sin{\left(x \right)}$$

Simplify

The graph

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Identical expressions

Derivative of sin^2x/cosx

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Piecewise:

The solution