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

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

Function f() - derivative -N order at the point
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You have entered [src] 
   ___________________
  /    2         2    
\/  cos (x) - sin (x)
−sin⁡2(x)+cos⁡2(x)\sqrt{- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}} 
  /   ___________________\
d |  /    2         2    |
--\\/  cos (x) - sin (x) /
dx
ddx−sin⁡2(x)+cos⁡2(x)\frac{d}{d x} \sqrt{- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}
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Detail solution

  1. Let u=−sin⁡2(x)+cos⁡2(x)u = - \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}.

  2. Apply the power rule: u\sqrt{u} goes to 12u\frac{1}{2 \sqrt{u}}

  3. Then, apply the chain rule. Multiply by ddx(−sin⁡2(x)+cos⁡2(x))\frac{d}{d x} \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right):

    1. Differentiate −sin⁡2(x)+cos⁡2(x)- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} term by term:

      1. Let u=cos⁡(x)u = \cos{\left(x \right)}.

      2. Apply the power rule: u2u^{2} goes to 2u2 u

      3. Then, apply the chain rule. Multiply by ddxcos⁡(x)\frac{d}{d x} \cos{\left(x \right)}:

        1. The derivative of cosine is negative sine:

          ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

        The result of the chain rule is:

        −2sin⁡(x)cos⁡(x)- 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⁡(x)u = \sin{\left(x \right)}.

        2. Apply the power rule: u2u^{2} goes to 2u2 u

        3. Then, apply the chain rule. Multiply by ddxsin⁡(x)\frac{d}{d x} \sin{\left(x \right)}:

          1. The derivative of sine is cosine:

            ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

          The result of the chain rule is:

          2sin⁡(x)cos⁡(x)2 \sin{\left(x \right)} \cos{\left(x \right)}

        So, the result is: −2sin⁡(x)cos⁡(x)- 2 \sin{\left(x \right)} \cos{\left(x \right)}

      The result is: −4sin⁡(x)cos⁡(x)- 4 \sin{\left(x \right)} \cos{\left(x \right)}

    The result of the chain rule is:

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

  4. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-10105-5
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)
−2sin⁡(x)cos⁡(x)−sin⁡2(x)+cos⁡2(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)
2(sin⁡2(x)−cos⁡2(x)−2sin⁡2(x)cos⁡2(x)−sin⁡2(x)+cos⁡2(x))−sin⁡2(x)+cos⁡2(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)/ /              
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                                 ___________________                              
                                /    2         2                                  
                              \/  cos (x) - sin (x)
4⋅(2+3sin⁡2(x)−sin⁡2(x)+cos⁡2(x)−3cos⁡2(x)−sin⁡2(x)+cos⁡2(x)−6sin⁡2(x)cos⁡2(x)(−sin⁡2(x)+cos⁡2(x))2)sin⁡(x)cos⁡(x)−sin⁡2(x)+cos⁡2(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
Derivative of sqrt(cos^2x-sin^2x)
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