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Derivative of:
Derivative of x^4*e^x
Derivative of x^2*sin(2*x-3)
Derivative of x^2/cosx
Derivative of sin(x)+cot(x)
Identical expressions
x^ two /cosx
x squared divide by co sinus of e of x
x to the power of two divide by co sinus of e of x
x2/cosx
x²/cosx
x to the power of 2/cosx
x^2 divide by cosx

The derivative of the function/ x^2/cosx

Derivative of x^2/cosx

The solution

You have entered [src]

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

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

x^2/cos(x)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{2}$ goes to $2 x$
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:

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

$\frac{x \left(x \tan{\left(x \right)} + 2\right)}{\cos{\left(x \right)}}$

The answer is:

$\frac{x \left(x \tan{\left(x \right)} + 2\right)}{\cos{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                    /         2   \       
                                  2 |    6*sin (x)|       
                                 x *|5 + ---------|*sin(x)
    /         2   \                 |        2    |       
    |    2*sin (x)|   6*sin(x)      \     cos (x) /       
6*x*|1 + ---------| + -------- + -------------------------
    |        2    |    cos(x)              cos(x)         
    \     cos (x) /                                       
----------------------------------------------------------
                          cos(x)

$$\frac{\frac{x^{2} \left(\frac{6 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 5\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + 6 x \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)}}}{\cos{\left(x \right)}}$$

Simplify

The graph

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Identical expressions

Derivative of x^2/cosx

The graph:

Piecewise:

The solution