Derivative of sin^n(x)

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   n   
sin (x)

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

d /   n   \
--\sin (x)/
dx

$$\frac{\partial}{\partial x} \sin^{n}{\left(x \right)}$$

Transform

Detail solution

Let $u = \sin{\left(x \right)}$ .
Apply the power rule: $u^{n}$ goes to $\frac{n u^{n}}{u}$
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:

$\frac{n \sin^{n}{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}}$
Now simplify:

$n \sin^{n - 1}{\left(x \right)} \cos{\left(x \right)}$

The answer is:

$n \sin^{n - 1}{\left(x \right)} \cos{\left(x \right)}$

The first derivative [src]

     n          
n*sin (x)*cos(x)
----------------
     sin(x)

$$\frac{n \sin^{n}{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}}$$

Simplify

The second derivative [src]

          /        2           2   \
     n    |     cos (x)   n*cos (x)|
n*sin (x)*|-1 - ------- + ---------|
          |        2          2    |
          \     sin (x)    sin (x) /

$$n \left(\frac{n \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - 1 - \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \sin^{n}{\left(x \right)}$$

Simplify

The third derivative [src]

          /               2       2    2             2   \       
     n    |          2*cos (x)   n *cos (x)   3*n*cos (x)|       
n*sin (x)*|2 - 3*n + --------- + ---------- - -----------|*cos(x)
          |              2           2             2     |       
          \           sin (x)     sin (x)       sin (x)  /       
-----------------------------------------------------------------
                              sin(x)

$$\frac{n \left(\frac{n^{2} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - 3 n - \frac{3 n \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 2 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \sin^{n}{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}}$$

Simplify

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Derivative of sin^n(x)

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