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The derivative of the function/ 3sinxcosx

Derivative of 3sinxcosx

The solution

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3*sin(x)*cos(x)

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

d                  
--(3*sin(x)*cos(x))
dx

$$\frac{d}{d x} 3 \sin{\left(x \right)} \cos{\left(x \right)}$$

Transform

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \cos{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  $g{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result is: $- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}$
So, the result is: $- 3 \sin^{2}{\left(x \right)} + 3 \cos^{2}{\left(x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

-12*cos(x)*sin(x)

$$- 12 \sin{\left(x \right)} \cos{\left(x \right)}$$

Simplify

The third derivative [src]

   /   2         2   \
12*\sin (x) - cos (x)/

$$12 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right)$$

Simplify

The graph

Derivative of 3sinxcosx