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Derivative of:
Derivative of e^(2*cos(t)^(2)-2*sin(t)^(2))
Derivative of cos(sqrt(x))
Derivative of 3*sqrt(x)
Derivative of 2x+1
Identical expressions
three *sin^2x*cosx
3 multiply by sinus of squared x multiply by co sinus of e of x
three multiply by sinus of squared x multiply by co sinus of e of x
3*sin2x*cosx
3*sin²x*cosx
3*sin to the power of 2x*cosx
3sin^2xcosx
3sin2xcosx

The derivative of the function/ 3*sin^2x*cosx

Derivative of 3sin^2xcosx

The solution

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     2          
3*sin (x)*cos(x)

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

(3*sin(x)^2)*cos(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = 3 \sin^{2}{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
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: $6 \sin{\left(x \right)} \cos{\left(x \right)}$
$g{\left(x \right)} = \cos{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of cosine is negative sine:
  
  $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
The result is: $- 3 \sin^{3}{\left(x \right)} + 6 \sin{\left(x \right)} \cos^{2}{\left(x \right)}$
Now simplify:

$- 9 \sin^{3}{\left(x \right)} + 6 \sin{\left(x \right)}$

The answer is:

$- 9 \sin^{3}{\left(x \right)} + 6 \sin{\left(x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /        2           2   \       
3*\- 20*cos (x) + 7*sin (x)/*sin(x)

$$3 \left(7 \sin^{2}{\left(x \right)} - 20 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}$$

Simplify