Derivative of z^2*sinz

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 2       
z *sin(z)

$$z^{2} \sin{\left(z \right)}$$

z^2*sin(z)

Detail solution

Apply the product rule:

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

$f{\left(z \right)} = z^{2}$ ; to find $\frac{d}{d z} f{\left(z \right)}$ :
1. Apply the power rule: $z^{2}$ goes to $2 z$
$g{\left(z \right)} = \sin{\left(z \right)}$ ; to find $\frac{d}{d z} g{\left(z \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d z} \sin{\left(z \right)} = \cos{\left(z \right)}$
The result is: $z^{2} \cos{\left(z \right)} + 2 z \sin{\left(z \right)}$
Now simplify:

$z \left(z \cos{\left(z \right)} + 2 \sin{\left(z \right)}\right)$

The answer is:

$z \left(z \cos{\left(z \right)} + 2 \sin{\left(z \right)}\right)$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 2                    
z *cos(z) + 2*z*sin(z)

$$z^{2} \cos{\left(z \right)} + 2 z \sin{\left(z \right)}$$

Simplify

The second derivative [src]

            2                    
2*sin(z) - z *sin(z) + 4*z*cos(z)

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

Simplify

The third derivative [src]

            2                    
6*cos(z) - z *cos(z) - 6*z*sin(z)

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

Simplify

The graph