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Derivative of:
Derivative of x^3/cosx
Derivative of (4-3*x)^7
Derivative of 3x²-2
Derivative of 2^x*sin(x)
Identical expressions
x^ three /cosx
x cubed divide by co sinus of e of x
x to the power of three divide by co sinus of e of x
x3/cosx
x³/cosx
x to the power of 3/cosx
x^3 divide by cosx

The derivative of the function/ x^3/cosx

Derivative of x^3/cosx

The solution

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

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

x^3/cos(x)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                            /         2   \       
                                          3 |    6*sin (x)|       
                                         x *|5 + ---------|*sin(x)
         /         2   \                    |        2    |       
       2 |    2*sin (x)|   18*x*sin(x)      \     cos (x) /       
6 + 9*x *|1 + ---------| + ----------- + -------------------------
         |        2    |      cos(x)               cos(x)         
         \     cos (x) /                                          
------------------------------------------------------------------
                              cos(x)

$$\frac{\frac{x^{3} \left(\frac{6 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 5\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + 9 x^{2} \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{18 x \sin{\left(x \right)}}{\cos{\left(x \right)}} + 6}{\cos{\left(x \right)}}$$

Simplify

The graph

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Identical expressions

Derivative of x^3/cosx

The graph:

Piecewise:

The solution