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x^3*cot(x)
The derivative of the function/ x^3*cot(x)

Derivative of x^3*cot(x)

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

You have entered [src] 
 3       
x *cot(x)
x3cot⁡(x)x^{3} \cot{\left(x \right)} 
x^3*cot(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)=x3f{\left(x \right)} = x^{3}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Apply the power rule: x3x^{3} goes to 3x23 x^{2}

    g(x)=cot⁡(x)g{\left(x \right)} = \cot{\left(x \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. There are multiple ways to do this derivative.

      Method #1

      1. Rewrite the function to be differentiated:

        cot⁡(x)=1tan⁡(x)\cot{\left(x \right)} = \frac{1}{\tan{\left(x \right)}}

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

      3. Apply the power rule: 1u\frac{1}{u} goes to −1u2- \frac{1}{u^{2}}

      4. 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:

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

      Method #2

      1. Rewrite the function to be differentiated:

        cot⁡(x)=cos⁡(x)sin⁡(x)\cot{\left(x \right)} = \frac{\cos{\left(x \right)}}{\sin{\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)=cos⁡(x)f{\left(x \right)} = \cos{\left(x \right)} and g(x)=sin⁡(x)g{\left(x \right)} = \sin{\left(x \right)}.

        To find ddxf(x)\frac{d}{d x} f{\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)}

        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)}

        Now plug in to the quotient rule:

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

    The result is: −x3(sin⁡2(x)+cos⁡2(x))cos⁡2(x)tan⁡2(x)+3x2cot⁡(x)- \frac{x^{3} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} + 3 x^{2} \cot{\left(x \right)}

  2. Now simplify:

    x2(−xsin⁡2(x)+3tan⁡(x))x^{2} \left(- \frac{x}{\sin^{2}{\left(x \right)}} + \frac{3}{\tan{\left(x \right)}}\right)

The answer is:

x2(−xsin⁡2(x)+3tan⁡(x))x^{2} \left(- \frac{x}{\sin^{2}{\left(x \right)}} + \frac{3}{\tan{\left(x \right)}}\right)

The graph
02468-8-6-4-2-1010-20000002000000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
 3 /        2   \      2       
x *\-1 - cot (x)/ + 3*x *cot(x)
x3(−cot⁡2(x)−1)+3x2cot⁡(x)x^{3} \left(- \cot^{2}{\left(x \right)} - 1\right) + 3 x^{2} \cot{\left(x \right)}
Simplify
The second derivative [src] 
    /               /       2   \    2 /       2   \       \
2*x*\3*cot(x) - 3*x*\1 + cot (x)/ + x *\1 + cot (x)/*cot(x)/
2x(x2(cot⁡2(x)+1)cot⁡(x)−3x(cot⁡2(x)+1)+3cot⁡(x))2 x \left(x^{2} \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} - 3 x \left(\cot^{2}{\left(x \right)} + 1\right) + 3 \cot{\left(x \right)}\right)
Simplify
The third derivative [src] 
  /               /       2   \    3 /       2   \ /         2   \      2 /       2   \       \
2*\3*cot(x) - 9*x*\1 + cot (x)/ - x *\1 + cot (x)/*\1 + 3*cot (x)/ + 9*x *\1 + cot (x)/*cot(x)/
2(−x3(cot⁡2(x)+1)(3cot⁡2(x)+1)+9x2(cot⁡2(x)+1)cot⁡(x)−9x(cot⁡2(x)+1)+3cot⁡(x))2 \left(- x^{3} \left(\cot^{2}{\left(x \right)} + 1\right) \left(3 \cot^{2}{\left(x \right)} + 1\right) + 9 x^{2} \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} - 9 x \left(\cot^{2}{\left(x \right)} + 1\right) + 3 \cot{\left(x \right)}\right)
Simplify
The graph
Derivative of x^3*cot(x)
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