Derivative of y=sin^nxcosnx

The solution

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

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

sin(x)^n*cos(n*x)

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = \sin^{n}{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \sin{\left(x \right)}$ .
2. Apply the power rule: $u^{n}$ goes to $\frac{n u^{n}}{u}$
3. 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:
  
  $\frac{n \sin^{n}{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}}$
$g{\left(x \right)} = \cos{\left(n x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = n x$ .
2. The derivative of cosine is negative sine:
  
  $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{\partial}{\partial x} n x$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $n$
  The result of the chain rule is:
  
  $- n \sin{\left(n x \right)}$
The result is: $- n \sin^{n}{\left(x \right)} \sin{\left(n x \right)} + \frac{n \sin^{n}{\left(x \right)} \cos{\left(x \right)} \cos{\left(n x \right)}}{\sin{\left(x \right)}}$
Now simplify:

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

The answer is:

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

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

          /                                                       /               2       2    2             2   \                                       \
          |                                                       |          2*cos (x)   n *cos (x)   3*n*cos (x)|                                       |
          |                                                       |2 - 3*n + --------- + ---------- - -----------|*cos(x)*cos(n*x)                       |
          |                  /       2           2   \            |              2           2             2     |                      2                |
     n    | 2                |    cos (x)   n*cos (x)|            \           sin (x)     sin (x)       sin (x)  /                   3*n *cos(x)*cos(n*x)|
n*sin (x)*|n *sin(n*x) + 3*n*|1 + ------- - ---------|*sin(n*x) + ---------------------------------------------------------------- - --------------------|
          |                  |       2          2    |                                         sin(x)                                       sin(x)       |
          \                  \    sin (x)    sin (x) /                                                                                                   /

$$n \left(n^{2} \sin{\left(n x \right)} - \frac{3 n^{2} \cos{\left(x \right)} \cos{\left(n x \right)}}{\sin{\left(x \right)}} + 3 n \left(- \frac{n \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \sin{\left(n x \right)} + \frac{\left(\frac{n^{2} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - 3 n - \frac{3 n \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 2 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)} \cos{\left(n x \right)}}{\sin{\left(x \right)}}\right) \sin^{n}{\left(x \right)}$$

Simplify

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Derivative of y=sin^nxcosnx

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Derivative of y=sin^n*x*cosnx

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The solution

Derivative of y=sin^nxcosnx