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(x^ two + four)/cosx
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(x2+4)/cosx
x2+4/cosx
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x^2+4/cosx
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Similar expressions
(x^2-4)/cosx

The derivative of the function/ (x^2+4)/cosx

Derivative of (x^2+4)/cosx

The solution

You have entered [src]

 2    
x  + 4
------
cos(x)

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

(x^2 + 4)/cos(x)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x^{2} + 4$ and $g{\left(x \right)} = \cos{\left(x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x^{2} + 4$ term by term:
  1. The derivative of the constant $4$ is zero.
  2. Apply the power rule: $x^{2}$ goes to $2 x$
  The result is: $2 x$
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:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                          /         2   \       
                                 /     2\ |    6*sin (x)|       
                                 \4 + 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)

$$\frac{6 x \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{\left(x^{2} + 4\right) \left(\frac{6 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 5\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)}}}{\cos{\left(x \right)}}$$

Simplify

The graph

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Derivative of (x^2+4)/cosx

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The solution