Derivative of sin(x)^10

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   10   
sin  (x)

$$\sin^{10}{\left(x \right)}$$

sin(x)^10

Detail solution

Let $u = \sin{\left(x \right)}$ .
Apply the power rule: $u^{10}$ goes to $10 u^{9}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
The result of the chain rule is:

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

The answer is:

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

The graph

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The first derivative [src]

      9          
10*sin (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)/

$$10 \left(- \sin^{2}{\left(x \right)} + 9 \cos^{2}{\left(x \right)}\right) \sin^{8}{\left(x \right)}$$

Simplify

The third derivative [src]

      7    /       2            2   \       
40*sin (x)*\- 7*sin (x) + 18*cos (x)/*cos(x)

$$40 \left(- 7 \sin^{2}{\left(x \right)} + 18 \cos^{2}{\left(x \right)}\right) \sin^{7}{\left(x \right)} \cos{\left(x \right)}$$

Simplify

4-я производная [src]

      6    /     4             4             2       2   \
40*sin (x)*\7*sin (x) + 126*cos (x) - 117*cos (x)*sin (x)/

$$40 \left(7 \sin^{4}{\left(x \right)} - 117 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} + 126 \cos^{4}{\left(x \right)}\right) \sin^{6}{\left(x \right)}$$

Simplify

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Derivative of sin(x)^10

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The solution