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x^2*tan(x)

Derivative of x^2*tan(x)

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

You have entered [src]
 2       
x *tan(x)
x2tan(x)x^{2} \tan{\left(x \right)}
x^2*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)=x2f{\left(x \right)} = x^{2}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

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

      sin2(x)+cos2(x)cos2(x)\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

    The result is: x2(sin2(x)+cos2(x))cos2(x)+2xtan(x)\frac{x^{2} \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:

    x(x+sin(2x))cos2(x)\frac{x \left(x + \sin{\left(2 x \right)}\right)}{\cos^{2}{\left(x \right)}}


The answer is:

x(x+sin(2x))cos2(x)\frac{x \left(x + \sin{\left(2 x \right)}\right)}{\cos^{2}{\left(x \right)}}

The graph
02468-8-6-4-2-1010-2000020000
The first derivative [src]
 2 /       2   \             
x *\1 + tan (x)/ + 2*x*tan(x)
x2(tan2(x)+1)+2xtan(x)x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) + 2 x \tan{\left(x \right)}
The second derivative [src]
  /    /       2   \    2 /       2   \                \
2*\2*x*\1 + tan (x)/ + x *\1 + tan (x)/*tan(x) + tan(x)/
2(x2(tan2(x)+1)tan(x)+2x(tan2(x)+1)+tan(x))2 \left(x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 2 x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right)
The third derivative [src]
  /         2       2 /       2   \ /         2   \       /       2   \       \
2*\3 + 3*tan (x) + x *\1 + tan (x)/*\1 + 3*tan (x)/ + 6*x*\1 + tan (x)/*tan(x)/
2(x2(tan2(x)+1)(3tan2(x)+1)+6x(tan2(x)+1)tan(x)+3tan2(x)+3)2 \left(x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) + 6 x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 3 \tan^{2}{\left(x \right)} + 3\right)
The graph
Derivative of x^2*tan(x)