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Identical expressions
y=sin^23xcos^32x
y equally sinus of squared 3x co sinus of e of cubed 2x
y=sin23xcos32x
y=sin²3xcos³2x
y=sin to the power of 23xcos to the power of 32x
Similar expressions
y=sin^23*xcos^32*x

The derivative of the function/ y=sin^23xcos^32x

Derivative of y=sin^23xcos^32x

The solution

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   23       32   
sin  (x)*cos  (x)

$$\sin^{23}{\left(x \right)} \cos^{32}{\left(x \right)}$$

d /   23       32   \
--\sin  (x)*cos  (x)/
dx

$$\frac{d}{d x} \sin^{23}{\left(x \right)} \cos^{32}{\left(x \right)}$$

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \sin^{23}{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \sin{\left(x \right)}$ .
2. Apply the power rule: $u^{23}$ goes to $23 u^{22}$
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:
  
  $23 \sin^{22}{\left(x \right)} \cos{\left(x \right)}$
$g{\left(x \right)} = \cos^{32}{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \cos{\left(x \right)}$ .
2. Apply the power rule: $u^{32}$ goes to $32 u^{31}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  The result of the chain rule is:
  
  $- 32 \sin{\left(x \right)} \cos^{31}{\left(x \right)}$
The result is: $- 32 \sin^{24}{\left(x \right)} \cos^{31}{\left(x \right)} + 23 \sin^{22}{\left(x \right)} \cos^{33}{\left(x \right)}$
Now simplify:

$\left(23 - 55 \sin^{2}{\left(x \right)}\right) \sin^{22}{\left(x \right)} \cos^{31}{\left(x \right)}$

The answer is:

$\left(23 - 55 \sin^{2}{\left(x \right)}\right) \sin^{22}{\left(x \right)} \cos^{31}{\left(x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        31       24            33       22   
- 32*cos  (x)*sin  (x) + 23*cos  (x)*sin  (x)

$$- 32 \sin^{24}{\left(x \right)} \cos^{31}{\left(x \right)} + 23 \sin^{22}{\left(x \right)} \cos^{33}{\left(x \right)}$$

Simplify

The second derivative [src]

   30       21    /          2       2            2    /   2            2   \         2    /     2            2   \\
cos  (x)*sin  (x)*\- 1472*cos (x)*sin (x) - 23*cos (x)*\sin (x) - 22*cos (x)/ + 32*sin (x)*\- cos (x) + 31*sin (x)//

$$\left(- 23 \left(\sin^{2}{\left(x \right)} - 22 \cos^{2}{\left(x \right)}\right) \cos^{2}{\left(x \right)} + 32 \cdot \left(31 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} - 1472 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}\right) \sin^{21}{\left(x \right)} \cos^{30}{\left(x \right)}$$

Simplify

The third derivative [src]

   29       20    /        4    /        2             2   \         4    /         2            2   \           2       2    /   2            2   \           2       2    /     2            2   \\
cos  (x)*sin  (x)*\- 64*sin (x)*\- 47*cos (x) + 465*sin (x)/ - 23*cos (x)*\- 462*cos (x) + 67*sin (x)/ + 2208*cos (x)*sin (x)*\sin (x) - 22*cos (x)/ + 2208*cos (x)*sin (x)*\- cos (x) + 31*sin (x)//

$$\left(2208 \left(\sin^{2}{\left(x \right)} - 22 \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} + 2208 \cdot \left(31 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} - 23 \cdot \left(67 \sin^{2}{\left(x \right)} - 462 \cos^{2}{\left(x \right)}\right) \cos^{4}{\left(x \right)} - 64 \cdot \left(465 \sin^{2}{\left(x \right)} - 47 \cos^{2}{\left(x \right)}\right) \sin^{4}{\left(x \right)}\right) \sin^{20}{\left(x \right)} \cos^{29}{\left(x \right)}$$

Simplify

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Derivative of y=sin^23xcos^32x

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The solution