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Derivative of (4-x^2)sinx

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/     2\       
\4 - x /*sin(x)

$$\left(4 - x^{2}\right) \sin{\left(x \right)}$$

d //     2\       \
--\\4 - x /*sin(x)/
dx

$$\frac{d}{d x} \left(4 - x^{2}\right) \sin{\left(x \right)}$$

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = 4 - x^{2}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $4 - x^{2}$ term by term:
  1. The derivative of the constant $4$ is zero.
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    So, the result is: $- 2 x$
  The result is: $- 2 x$
$g{\left(x \right)} = \sin{\left(x \right)}$ ; 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: $- 2 x \sin{\left(x \right)} + \left(4 - x^{2}\right) \cos{\left(x \right)}$
Now simplify:

$- 2 x \sin{\left(x \right)} - \left(x^{2} - 4\right) \cos{\left(x \right)}$

The answer is:

$- 2 x \sin{\left(x \right)} - \left(x^{2} - 4\right) \cos{\left(x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

/     2\                    
\4 - x /*cos(x) - 2*x*sin(x)

$$- 2 x \sin{\left(x \right)} + \left(4 - x^{2}\right) \cos{\left(x \right)}$$

Simplify

The second derivative [src]

            /      2\                    
-2*sin(x) + \-4 + x /*sin(x) - 4*x*cos(x)

$$- 4 x \cos{\left(x \right)} + \left(x^{2} - 4\right) \sin{\left(x \right)} - 2 \sin{\left(x \right)}$$

Simplify

The third derivative [src]

            /      2\                    
-6*cos(x) + \-4 + x /*cos(x) + 6*x*sin(x)

$$6 x \sin{\left(x \right)} + \left(x^{2} - 4\right) \cos{\left(x \right)} - 6 \cos{\left(x \right)}$$

Simplify

The graph