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The derivative of the function/ (2-sqrt(x))*tgx

Derivative of (2-sqrt(x))*tgx

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

You have entered [src] 
/      ___\       
\2 - \/ x /*tan(x)
(2−x)tan⁡(x)\left(2 - \sqrt{x}\right) \tan{\left(x \right)} 
(2 - sqrt(x))*tan(x)
Detail solution

  1. Apply the product rule:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}

    f(x)=2−xf{\left(x \right)} = 2 - \sqrt{x}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Differentiate 2−x2 - \sqrt{x} term by term:

      1. The derivative of the constant 22 is zero.

      2. The derivative of a constant times a function is the constant times the derivative of the function.

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

        So, the result is: −12x- \frac{1}{2 \sqrt{x}}

      The result is: −12x- \frac{1}{2 \sqrt{x}}

    g(x)=tan⁡(x)g{\left(x \right)} = \tan{\left(x \right)}; to find ddxg(x)\frac{d}{d x} g{\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 is: (2−x)(sin⁡2(x)+cos⁡2(x))cos⁡2(x)−tan⁡(x)2x\frac{\left(2 - \sqrt{x}\right) \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} - \frac{\tan{\left(x \right)}}{2 \sqrt{x}}

  2. Now simplify:

    2x−x−sin⁡(2x)4xcos⁡2(x)\frac{2 \sqrt{x} - x - \frac{\sin{\left(2 x \right)}}{4}}{\sqrt{x} \cos^{2}{\left(x \right)}}

The answer is:

2x−x−sin⁡(2x)4xcos⁡2(x)\frac{2 \sqrt{x} - x - \frac{\sin{\left(2 x \right)}}{4}}{\sqrt{x} \cos^{2}{\left(x \right)}}

The graph
02468-8-6-4-2-1010-500500
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The first derivative [src] 
/       2   \ /      ___\    tan(x)
\1 + tan (x)/*\2 - \/ x / - -------
                                ___
                            2*\/ x
(2−x)(tan⁡2(x)+1)−tan⁡(x)2x\left(2 - \sqrt{x}\right) \left(\tan^{2}{\left(x \right)} + 1\right) - \frac{\tan{\left(x \right)}}{2 \sqrt{x}}
Simplify
The second derivative [src] 
         2                                                  
  1 + tan (x)   tan(x)     /       2   \ /       ___\       
- ----------- + ------ - 2*\1 + tan (x)/*\-2 + \/ x /*tan(x)
       ___         3/2                                      
     \/ x       4*x
−2(x−2)(tan⁡2(x)+1)tan⁡(x)−tan⁡2(x)+1x+tan⁡(x)4x32- 2 \left(\sqrt{x} - 2\right) \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{x}} + \frac{\tan{\left(x \right)}}{4 x^{\frac{3}{2}}}
Simplify
The third derivative [src] 
               /       2   \     /       2   \                                                      
  3*tan(x)   3*\1 + tan (x)/   3*\1 + tan (x)/*tan(x)     /       2   \ /         2   \ /       ___\
- -------- + --------------- - ---------------------- - 2*\1 + tan (x)/*\1 + 3*tan (x)/*\-2 + \/ x /
      5/2            3/2                 ___                                                        
   8*x            4*x                  \/ x
−2(x−2)(tan⁡2(x)+1)(3tan⁡2(x)+1)−3(tan⁡2(x)+1)tan⁡(x)x+3(tan⁡2(x)+1)4x32−3tan⁡(x)8x52- 2 \left(\sqrt{x} - 2\right) \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{\sqrt{x}} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{4 x^{\frac{3}{2}}} - \frac{3 \tan{\left(x \right)}}{8 x^{\frac{5}{2}}}
Simplify
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