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Derivative of y=(x^2-7)cosx

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/ 2    \       
\x  - 7/*cos(x)

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

(x^2 - 7)*cos(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x^{2} - 7$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x^{2} - 7$ term by term:
  1. Apply the power rule: $x^{2}$ goes to $2 x$
  2. The derivative of the constant $-7$ is zero.
  The result is: $2 x$
$g{\left(x \right)} = \cos{\left(x \right)}$ ; 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: $2 x \cos{\left(x \right)} - \left(x^{2} - 7\right) \sin{\left(x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

           /      2\                    
2*cos(x) - \-7 + x /*cos(x) - 4*x*sin(x)

$$- 4 x \sin{\left(x \right)} - \left(x^{2} - 7\right) \cos{\left(x \right)} + 2 \cos{\left(x \right)}$$

Simplify

The third derivative [src]

            /      2\                    
-6*sin(x) + \-7 + x /*sin(x) - 6*x*cos(x)

$$- 6 x \cos{\left(x \right)} + \left(x^{2} - 7\right) \sin{\left(x \right)} - 6 \sin{\left(x \right)}$$

Simplify

The graph