Mister Exam

Other calculators

The derivative of the function/ tan(xsin(x))

Derivative of tan(xsin(x))

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
tan(x*sin(x))
tan⁡(xsin⁡(x))\tan{\left(x \sin{\left(x \right)} \right)} 
tan(x*sin(x))
Detail solution

  1. Rewrite the function to be differentiated:

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

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

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

    2. The derivative of sine is cosine:

      ddusin⁡(u)=cos⁡(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

    3. Then, apply the chain rule. Multiply by ddxxsin⁡(x)\frac{d}{d x} x \sin{\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)=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)=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: xcos⁡(x)+sin⁡(x)x \cos{\left(x \right)} + \sin{\left(x \right)}

      The result of the chain rule is:

      (xcos⁡(x)+sin⁡(x))cos⁡(xsin⁡(x))\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \cos{\left(x \sin{\left(x \right)} \right)}

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

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

    2. The derivative of cosine is negative sine:

      dducos⁡(u)=−sin⁡(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

    3. Then, apply the chain rule. Multiply by ddxxsin⁡(x)\frac{d}{d x} x \sin{\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)=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)=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: xcos⁡(x)+sin⁡(x)x \cos{\left(x \right)} + \sin{\left(x \right)}

      The result of the chain rule is:

      −(xcos⁡(x)+sin⁡(x))sin⁡(xsin⁡(x))- \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \sin{\left(x \right)} \right)}

    Now plug in to the quotient rule:

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

  3. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-50005000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
/       2          \                    
\1 + tan (x*sin(x))/*(x*cos(x) + sin(x))
(xcos⁡(x)+sin⁡(x))(tan⁡2(xsin⁡(x))+1)\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \left(\tan^{2}{\left(x \sin{\left(x \right)} \right)} + 1\right)
Simplify
The second derivative [src] 
/       2          \ /                                           2              \
\1 + tan (x*sin(x))/*\2*cos(x) - x*sin(x) + 2*(x*cos(x) + sin(x)) *tan(x*sin(x))/
(tan⁡2(xsin⁡(x))+1)(−xsin⁡(x)+2(xcos⁡(x)+sin⁡(x))2tan⁡(xsin⁡(x))+2cos⁡(x))\left(\tan^{2}{\left(x \sin{\left(x \right)} \right)} + 1\right) \left(- x \sin{\left(x \right)} + 2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2} \tan{\left(x \sin{\left(x \right)} \right)} + 2 \cos{\left(x \right)}\right)
Simplify
The third derivative [src] 
/       2          \ /                                            3 /       2          \                        3    2                                                                       \
\1 + tan (x*sin(x))/*\-3*sin(x) - x*cos(x) + 2*(x*cos(x) + sin(x)) *\1 + tan (x*sin(x))/ + 4*(x*cos(x) + sin(x)) *tan (x*sin(x)) - 6*(-2*cos(x) + x*sin(x))*(x*cos(x) + sin(x))*tan(x*sin(x))/
(tan⁡2(xsin⁡(x))+1)(−xcos⁡(x)−6(xsin⁡(x)−2cos⁡(x))(xcos⁡(x)+sin⁡(x))tan⁡(xsin⁡(x))+2(xcos⁡(x)+sin⁡(x))3(tan⁡2(xsin⁡(x))+1)+4(xcos⁡(x)+sin⁡(x))3tan⁡2(xsin⁡(x))−3sin⁡(x))\left(\tan^{2}{\left(x \sin{\left(x \right)} \right)} + 1\right) \left(- x \cos{\left(x \right)} - 6 \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \tan{\left(x \sin{\left(x \right)} \right)} + 2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{3} \left(\tan^{2}{\left(x \sin{\left(x \right)} \right)} + 1\right) + 4 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{3} \tan^{2}{\left(x \sin{\left(x \right)} \right)} - 3 \sin{\left(x \right)}\right)
Simplify
    © Mister Exam – Calculator