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The derivative of the function/ x^2sinxtanx

Derivative of x^2sinxtanx

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

You have entered [src] 
 2              
x *sin(x)*tan(x)
x2sin⁡(x)tan⁡(x)x^{2} \sin{\left(x \right)} \tan{\left(x \right)} 
(x^2*sin(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)=x2sin⁡(x)f{\left(x \right)} = x^{2} \sin{\left(x \right)}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    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)=sin⁡(x)g{\left(x \right)} = \sin{\left(x \right)}; to find ddxg(x)\frac{d}{d x} g{\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 is: x2cos⁡(x)+2xsin⁡(x)x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)}

    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: x2(sin⁡2(x)+cos⁡2(x))sin⁡(x)cos⁡2(x)+(x2cos⁡(x)+2xsin⁡(x))tan⁡(x)\frac{x^{2} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \left(x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)}\right) \tan{\left(x \right)}

  2. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-5000050000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
/ 2                    \           2 /       2   \       
\x *cos(x) + 2*x*sin(x)/*tan(x) + x *\1 + tan (x)/*sin(x)
x2(tan⁡2(x)+1)sin⁡(x)+(x2cos⁡(x)+2xsin⁡(x))tan⁡(x)x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} + \left(x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)}\right) \tan{\left(x \right)}
Simplify
The second derivative [src] 
/            2                    \              /       2   \                            2 /       2   \              
\2*sin(x) - x *sin(x) + 4*x*cos(x)/*tan(x) + 2*x*\1 + tan (x)/*(2*sin(x) + x*cos(x)) + 2*x *\1 + tan (x)/*sin(x)*tan(x)
2x2(tan⁡2(x)+1)sin⁡(x)tan⁡(x)+2x(xcos⁡(x)+2sin⁡(x))(tan⁡2(x)+1)+(−x2sin⁡(x)+4xcos⁡(x)+2sin⁡(x))tan⁡(x)2 x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \tan{\left(x \right)} + 2 x \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) + \left(- x^{2} \sin{\left(x \right)} + 4 x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \tan{\left(x \right)}
Simplify
The third derivative [src] 
  /             2                    \            /       2   \ /            2                    \      2 /       2   \ /         2   \              /       2   \                             
- \-6*cos(x) + x *cos(x) + 6*x*sin(x)/*tan(x) + 3*\1 + tan (x)/*\2*sin(x) - x *sin(x) + 4*x*cos(x)/ + 2*x *\1 + tan (x)/*\1 + 3*tan (x)/*sin(x) + 6*x*\1 + tan (x)/*(2*sin(x) + x*cos(x))*tan(x)
2x2(tan⁡2(x)+1)(3tan⁡2(x)+1)sin⁡(x)+6x(xcos⁡(x)+2sin⁡(x))(tan⁡2(x)+1)tan⁡(x)+3(tan⁡2(x)+1)(−x2sin⁡(x)+4xcos⁡(x)+2sin⁡(x))−(x2cos⁡(x)+6xsin⁡(x)−6cos⁡(x))tan⁡(x)2 x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} + 6 x \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 3 \left(\tan^{2}{\left(x \right)} + 1\right) \left(- x^{2} \sin{\left(x \right)} + 4 x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) - \left(x^{2} \cos{\left(x \right)} + 6 x \sin{\left(x \right)} - 6 \cos{\left(x \right)}\right) \tan{\left(x \right)}
Simplify
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