Mister Exam

Other calculators

The derivative of the function/ xtg^3x

Derivative of xtg^3x

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
     3   
x*tan (x)
xtan⁡3(x)x \tan^{3}{\left(x \right)} 
x*tan(x)^3
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)=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⁡3(x)g{\left(x \right)} = \tan^{3}{\left(x \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

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

    2. Apply the power rule: u3u^{3} goes to 3u23 u^{2}

    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:

      3(sin⁡2(x)+cos⁡2(x))tan⁡2(x)cos⁡2(x)\frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

    The result is: 3x(sin⁡2(x)+cos⁡2(x))tan⁡2(x)cos⁡2(x)+tan⁡3(x)\frac{3 x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \tan^{3}{\left(x \right)}

  2. Now simplify:

    3xsin⁡2(x)cos⁡4(x)+tan⁡3(x)\frac{3 x \sin^{2}{\left(x \right)}}{\cos^{4}{\left(x \right)}} + \tan^{3}{\left(x \right)}

The answer is:

3xsin⁡2(x)cos⁡4(x)+tan⁡3(x)\frac{3 x \sin^{2}{\left(x \right)}}{\cos^{4}{\left(x \right)}} + \tan^{3}{\left(x \right)}

The graph
02468-8-6-4-2-1010-2000000020000000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   3           2    /         2   \
tan (x) + x*tan (x)*\3 + 3*tan (x)/
x(3tan⁡2(x)+3)tan⁡2(x)+tan⁡3(x)x \left(3 \tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)} + \tan^{3}{\left(x \right)}
Simplify
The second derivative [src] 
  /       2   \ /  /         2   \         \       
6*\1 + tan (x)/*\x*\1 + 2*tan (x)/ + tan(x)/*tan(x)
6(x(2tan⁡2(x)+1)+tan⁡(x))(tan⁡2(x)+1)tan⁡(x)6 \left(x \left(2 \tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}
Simplify
The third derivative [src] 
                /  /             2                                      \                           \
  /       2   \ |  |/       2   \         4           2    /       2   \|     /         2   \       |
6*\1 + tan (x)/*\x*\\1 + tan (x)/  + 2*tan (x) + 7*tan (x)*\1 + tan (x)// + 3*\1 + 2*tan (x)/*tan(x)/
6(x((tan⁡2(x)+1)2+7(tan⁡2(x)+1)tan⁡2(x)+2tan⁡4(x))+3(2tan⁡2(x)+1)tan⁡(x))(tan⁡2(x)+1)6 \left(x \left(\left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 7 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 2 \tan^{4}{\left(x \right)}\right) + 3 \left(2 \tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right)
Simplify
    © Mister Exam – Calculator