Derivative of 3sin^2x-sin^3x

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     2         3   
3*sin (x) - sin (x)

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

d /     2         3   \
--\3*sin (x) - sin (x)/
dx

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

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Detail solution

Differentiate $- \sin^{3}{\left(x \right)} + 3 \sin^{2}{\left(x \right)}$ term by term:
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \sin{\left(x \right)}$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  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:
    
    $2 \sin{\left(x \right)} \cos{\left(x \right)}$
  So, the result is: $6 \sin{\left(x \right)} \cos{\left(x \right)}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \sin{\left(x \right)}$ .
  2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
  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:
    
    $3 \sin^{2}{\left(x \right)} \cos{\left(x \right)}$
  So, the result is: $- 3 \sin^{2}{\left(x \right)} \cos{\left(x \right)}$
The result is: $- 3 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + 6 \sin{\left(x \right)} \cos{\left(x \right)}$
Now simplify:

$3 \cdot \left(2 - \sin{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}$

The answer is:

$3 \cdot \left(2 - \sin{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2                            
- 3*sin (x)*cos(x) + 6*cos(x)*sin(x)

$$- 3 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + 6 \sin{\left(x \right)} \cos{\left(x \right)}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                 2           2   \       
3*\-8*sin(x) - 2*cos (x) + 7*sin (x)/*cos(x)

$$3 \cdot \left(7 \sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)} - 8 \sin{\left(x \right)}\right) \cos{\left(x \right)}$$

Simplify

The graph

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Derivative of 3sin^2x-sin^3x

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The solution