Derivative of y=(sinx-3x)(x+1)

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(sin(x) - 3*x)*(x + 1)

$$\left(- 3 x + \sin{\left(x \right)}\right) \left(x + 1\right)$$

(sin(x) - 3*x)*(x + 1)

Detail solution

Apply the product rule:

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

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

$- 3 x + \left(x + 1\right) \left(\cos{\left(x \right)} - 3\right) + \sin{\left(x \right)}$

The answer is:

$- 3 x + \left(x + 1\right) \left(\cos{\left(x \right)} - 3\right) + \sin{\left(x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

-3*x + (-3 + cos(x))*(x + 1) + sin(x)

$$- 3 x + \left(x + 1\right) \left(\cos{\left(x \right)} - 3\right) + \sin{\left(x \right)}$$

Simplify

The second derivative [src]

-6 + 2*cos(x) - (1 + x)*sin(x)

$$- \left(x + 1\right) \sin{\left(x \right)} + 2 \cos{\left(x \right)} - 6$$

Simplify

The third derivative [src]

-(3*sin(x) + (1 + x)*cos(x))

$$- (\left(x + 1\right) \cos{\left(x \right)} + 3 \sin{\left(x \right)})$$

Simplify

The graph