Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of y=√4-x^2
Derivative of x^2/(x+2)
Derivative of x^2*tgx
Derivative of sin3x^2
Identical expressions
-sqrt(cos^ two (x)+ two)
minus square root of ( co sinus of e of squared (x) plus 2)
minus square root of ( co sinus of e of to the power of two (x) plus two)
-√(cos^2(x)+2)
-sqrt(cos2(x)+2)
-sqrtcos2x+2
-sqrt(cos²(x)+2)
-sqrt(cos to the power of 2(x)+2)
-sqrtcos^2x+2
Similar expressions
-sqrt(cos^2(x)-2)
sqrt(cos^2(x)+2)

The derivative of the function/ -sqrt(cos^2(x)+2)

Derivative of -sqrt(cos^2(x)+2)

The solution

You have entered [src]

    _____________
   /    2        
-\/  cos (x) + 2

$$- \sqrt{\cos^{2}{\left(x \right)} + 2}$$

-sqrt(cos(x)^2 + 2)

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \cos^{2}{\left(x \right)} + 2$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\cos^{2}{\left(x \right)} + 2\right)$ :
  1. Differentiate $\cos^{2}{\left(x \right)} + 2$ term by term:
    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:
      
      $- 2 \sin{\left(x \right)} \cos{\left(x \right)}$
    4. The derivative of the constant $2$ is zero.
    The result is: $- 2 \sin{\left(x \right)} \cos{\left(x \right)}$
  The result of the chain rule is:
  
  $- \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sqrt{\cos^{2}{\left(x \right)} + 2}}$
So, the result is: $\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sqrt{\cos^{2}{\left(x \right)} + 2}}$
Now simplify:

$\frac{\sqrt{2} \sin{\left(2 x \right)}}{2 \sqrt{\cos{\left(2 x \right)} + 5}}$

The answer is:

$\frac{\sqrt{2} \sin{\left(2 x \right)}}{2 \sqrt{\cos{\left(2 x \right)} + 5}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of -sqrt(cos^2(x)+2)

The graph:

Piecewise:

The solution