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sin(x)^(9)
The derivative of the function/ sin(x)^(9)

Derivative of sin(x)^(9)

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

You have entered [src] 
   9   
sin (x)
sin⁡9(x)\sin^{9}{\left(x \right)} 
sin(x)^9
Detail solution

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

  2. Apply the power rule: u9u^{9} goes to 9u89 u^{8}

  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:

    9sin⁡8(x)cos⁡(x)9 \sin^{8}{\left(x \right)} \cos{\left(x \right)}

The answer is:

9sin⁡8(x)cos⁡(x)9 \sin^{8}{\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] 
     8          
9*sin (x)*cos(x)
9sin⁡8(x)cos⁡(x)9 \sin^{8}{\left(x \right)} \cos{\left(x \right)}
Simplify
The second derivative [src] 
     7    /     2           2   \
9*sin (x)*\- sin (x) + 8*cos (x)/
9(−sin⁡2(x)+8cos⁡2(x))sin⁡7(x)9 \left(- \sin^{2}{\left(x \right)} + 8 \cos^{2}{\left(x \right)}\right) \sin^{7}{\left(x \right)}
Simplify
The third derivative [src] 
     6    /        2            2   \       
9*sin (x)*\- 25*sin (x) + 56*cos (x)/*cos(x)
9(−25sin⁡2(x)+56cos⁡2(x))sin⁡6(x)cos⁡(x)9 \left(- 25 \sin^{2}{\left(x \right)} + 56 \cos^{2}{\left(x \right)}\right) \sin^{6}{\left(x \right)} \cos{\left(x \right)}
Simplify
The graph
Derivative of sin(x)^(9)
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