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sqrttan(x)^(2)
The derivative of the function/ sqrttan(x)^(2)

Derivative of sqrttan(x)^(2)

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
          2
  ________ 
\/ tan(x)
(tan⁡(x))2\left(\sqrt{\tan{\left(x \right)}}\right)^{2} 
  /          2\
d |  ________ |
--\\/ tan(x)  /
dx
ddx(tan⁡(x))2\frac{d}{d x} \left(\sqrt{\tan{\left(x \right)}}\right)^{2}
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Detail solution

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

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

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

    1. Let u=tan⁡(x)u = \tan{\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 ddxtan⁡(x)\frac{d}{d x} \tan{\left(x \right)}:

      1. Rewrite the function to be differentiated:

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

      2. Apply the quotient rule, which is:

        ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=sin⁡(x)f{\left(x \right)} = \sin{\left(x \right)} and g(x)=cos⁡(x)g{\left(x \right)} = \cos{\left(x \right)}.

        To find ddxf(x)\frac{d}{d x} f{\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)}

        To find ddxg(x)\frac{d}{d x} g{\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)}

        Now plug in to the quotient rule:

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

      The result of the chain rule is:

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

    The result of the chain rule is:

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

  4. Now simplify:

    1cos⁡2(x)\frac{1}{\cos^{2}{\left(x \right)}}

The answer is:

1cos⁡2(x)\frac{1}{\cos^{2}{\left(x \right)}}

The graph
02468-8-6-4-2-1010-10001000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
  /       2   \       
  |1   tan (x)|       
2*|- + -------|*tan(x)
  \2      2   /       
----------------------
        tan(x)
2(tan⁡2(x)2+12)tan⁡(x)tan⁡(x)\frac{2 \left(\frac{\tan^{2}{\left(x \right)}}{2} + \frac{1}{2}\right) \tan{\left(x \right)}}{\tan{\left(x \right)}}
Simplify
The second derivative [src] 
  /       2   \       
2*\1 + tan (x)/*tan(x)
2(tan⁡2(x)+1)tan⁡(x)2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}
Simplify
The third derivative [src] 
  /       2   \ /         2   \
2*\1 + tan (x)/*\1 + 3*tan (x)/
2(tan⁡2(x)+1)(3tan⁡2(x)+1)2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right)
Simplify
The graph
Derivative of sqrttan(x)^(2)
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