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The derivative of the function/ sin(x)^10

Derivative of sin(x)^10

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

from to

Piecewise:

The solution

You have entered [src] 
   10   
sin  (x)
sin⁡10(x)\sin^{10}{\left(x \right)} 
sin(x)^10
Detail solution

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

  2. Apply the power rule: u10u^{10} goes to 10u910 u^{9}

  3. Then, apply the chain rule. Multiply by ddxsin⁡(x)\frac{d}{d x} \sin{\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 of the chain rule is:

    10sin⁡9(x)cos⁡(x)10 \sin^{9}{\left(x \right)} \cos{\left(x \right)}

The answer is:

10sin⁡9(x)cos⁡(x)10 \sin^{9}{\left(x \right)} \cos{\left(x \right)}

The graph
02468-8-6-4-2-10105-5
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
      9          
10*sin (x)*cos(x)
10sin⁡9(x)cos⁡(x)10 \sin^{9}{\left(x \right)} \cos{\left(x \right)}
Simplify
The second derivative [src] 
      8    /     2           2   \
10*sin (x)*\- sin (x) + 9*cos (x)/
10(−sin⁡2(x)+9cos⁡2(x))sin⁡8(x)10 \left(- \sin^{2}{\left(x \right)} + 9 \cos^{2}{\left(x \right)}\right) \sin^{8}{\left(x \right)}
Simplify
The third derivative [src] 
      7    /       2            2   \       
40*sin (x)*\- 7*sin (x) + 18*cos (x)/*cos(x)
40(−7sin⁡2(x)+18cos⁡2(x))sin⁡7(x)cos⁡(x)40 \left(- 7 \sin^{2}{\left(x \right)} + 18 \cos^{2}{\left(x \right)}\right) \sin^{7}{\left(x \right)} \cos{\left(x \right)}
Simplify
4-я производная [src] 
      6    /     4             4             2       2   \
40*sin (x)*\7*sin (x) + 126*cos (x) - 117*cos (x)*sin (x)/
40(7sin⁡4(x)−117sin⁡2(x)cos⁡2(x)+126cos⁡4(x))sin⁡6(x)40 \left(7 \sin^{4}{\left(x \right)} - 117 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} + 126 \cos^{4}{\left(x \right)}\right) \sin^{6}{\left(x \right)}
Simplify
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