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Derivative of:
Derivative of 2/x^2
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Derivative of (x+2)^2
Integral of d{x}:
sinx^2/cosx^2
Identical expressions
sinx^ two /cosx^ two
sinus of x squared divide by co sinus of e of x squared
sinus of x to the power of two divide by co sinus of e of x to the power of two
sinx2/cosx2
sinx²/cosx²
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sinx^2 divide by cosx^2

The derivative of the function/ sinx^2/cosx^2

Derivative of sinx^2/cosx^2

The solution

You have entered [src]

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

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

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

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

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                                                  /         2   \\       
  |                                             2    |    3*sin (x)||       
  |                                        2*sin (x)*|2 + ---------||       
  |      /   2         2   \        2                |        2    ||       
  |    3*\sin (x) - cos (x)/   9*sin (x)             \     cos (x) /|       
4*|1 - --------------------- + --------- + -------------------------|*sin(x)
  |              2                 2                   2            |       
  \           cos (x)           cos (x)             cos (x)         /       
----------------------------------------------------------------------------
                                   cos(x)

$$\frac{4 \cdot \left(\frac{2 \cdot \left(\frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{3 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \frac{9 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \sin{\left(x \right)}}{\cos{\left(x \right)}}$$

Simplify

The graph

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Identical expressions

Derivative of sinx^2/cosx^2

The graph:

Piecewise:

The solution