Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of sh(x)
Derivative of x^3-2x^2-5
Derivative of sqrt(cos^2x-sin^2x)
Derivative of 5x^6
Integral of d{x}:
sqrt(cos^2x-sin^2x)
Identical expressions
sqrt(cos^2x-sin^2x)
square root of ( co sinus of e of squared x minus sinus of squared x)
√(cos^2x-sin^2x)
sqrt(cos2x-sin2x)
sqrtcos2x-sin2x
sqrt(cos²x-sin²x)
sqrt(cos to the power of 2x-sin to the power of 2x)
sqrtcos^2x-sin^2x
Similar expressions
sqrt(cos^2x+sin^2x)

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

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

The solution

You have entered [src]

   ___________________
  /    2         2    
\/  cos (x) - sin (x)

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

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

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

Transform

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   -2*cos(x)*sin(x)   
----------------------
   ___________________
  /    2         2    
\/  cos (x) - sin (x)

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

Simplify

The second derivative [src]

  /                         2       2   \
  |   2         2      2*cos (x)*sin (x)|
2*|sin (x) - cos (x) - -----------------|
  |                       2         2   |
  \                    cos (x) - sin (x)/
-----------------------------------------
             ___________________         
            /    2         2             
          \/  cos (x) - sin (x)

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

Simplify

The third derivative [src]

  /             2                   2                2       2     \              
  |        3*cos (x)           3*sin (x)        6*cos (x)*sin (x)  |              
4*|2 - ----------------- + ----------------- - --------------------|*cos(x)*sin(x)
  |       2         2         2         2                         2|              
  |    cos (x) - sin (x)   cos (x) - sin (x)   /   2         2   \ |              
  \                                            \cos (x) - sin (x)/ /              
----------------------------------------------------------------------------------
                                 ___________________                              
                                /    2         2                                  
                              \/  cos (x) - sin (x)

$$\frac{4 \cdot \left(2 + \frac{3 \sin^{2}{\left(x \right)}}{- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}} - \frac{3 \cos^{2}{\left(x \right)}}{- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}} - \frac{6 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)^{2}}\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\sqrt{- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}}$$

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

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

The graph:

Piecewise:

The solution