Derivative of y=(x-8)(sinx)

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(x - 8)*sin(x)

$$\left(x - 8\right) \sin{\left(x \right)}$$

(x - 8)*sin(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x - 8$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x - 8$ term by term:
  1. Apply the power rule: $x$ goes to $1$
  2. The derivative of the constant $-8$ is zero.
  The result is: $1$
$g{\left(x \right)} = \sin{\left(x \right)}$ ; 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: $\left(x - 8\right) \cos{\left(x \right)} + \sin{\left(x \right)}$
Now simplify:

$\left(x - 8\right) \cos{\left(x \right)} + \sin{\left(x \right)}$

The answer is:

$\left(x - 8\right) \cos{\left(x \right)} + \sin{\left(x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

(x - 8)*cos(x) + sin(x)

$$\left(x - 8\right) \cos{\left(x \right)} + \sin{\left(x \right)}$$

Simplify

The second derivative [src]

2*cos(x) - (-8 + x)*sin(x)

$$- \left(x - 8\right) \sin{\left(x \right)} + 2 \cos{\left(x \right)}$$

Simplify

The third derivative [src]

-(3*sin(x) + (-8 + x)*cos(x))

$$- (\left(x - 8\right) \cos{\left(x \right)} + 3 \sin{\left(x \right)})$$

Simplify

The graph