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xtgx/1+x²
The derivative of the function/ xtgx/1+x²

Derivative of xtgx/1+x²

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

You have entered [src] 
x*tan(x)    2
-------- + x 
   1
x2+xtan⁡(x)1x^{2} + \frac{x \tan{\left(x \right)}}{1} 
d /x*tan(x)    2\
--|-------- + x |
dx\   1         /
ddx(x2+xtan⁡(x)1)\frac{d}{d x} \left(x^{2} + \frac{x \tan{\left(x \right)}}{1}\right)
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Detail solution

  1. Differentiate x2+xtan⁡(x)1x^{2} + \frac{x \tan{\left(x \right)}}{1} term by term:

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

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

        1. Apply the power rule: xx goes to 11

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

      So, the result is: x(sin⁡2(x)+cos⁡2(x))cos⁡2(x)+tan⁡(x)\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}

    2. Apply the power rule: x2x^{2} goes to 2x2 x

    The result is: x(sin⁡2(x)+cos⁡2(x))cos⁡2(x)+2x+tan⁡(x)\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + 2 x + \tan{\left(x \right)}

  2. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-1000010000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
        /       2   \         
2*x + x*\1 + tan (x)/ + tan(x)
x(tan⁡2(x)+1)+2x+tan⁡(x)x \left(\tan^{2}{\left(x \right)} + 1\right) + 2 x + \tan{\left(x \right)}
Simplify
The second derivative [src] 
  /       2        /       2   \       \
2*\2 + tan (x) + x*\1 + tan (x)/*tan(x)/
2(x(tan⁡2(x)+1)tan⁡(x)+tan⁡2(x)+2)2 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \tan^{2}{\left(x \right)} + 2\right)
Simplify
The third derivative [src] 
  /       2   \ /             /       2   \          2   \
2*\1 + tan (x)/*\3*tan(x) + x*\1 + tan (x)/ + 2*x*tan (x)/
2(tan⁡2(x)+1)(x(tan⁡2(x)+1)+2xtan⁡2(x)+3tan⁡(x))2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + 2 x \tan^{2}{\left(x \right)} + 3 \tan{\left(x \right)}\right)
Simplify
The graph
Derivative of xtgx/1+x²
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