Derivative of f(x)=(x-1)2sinx

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(x - 1)*2*sin(x)

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

((x - 1)*2)*sin(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = 2 \left(x - 1\right)$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  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: $1$
  So, the result is: $2$
$g{\left(x \right)} = \sin{\left(x \right)}$ ; 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: $2 \left(x - 1\right) \cos{\left(x \right)} + 2 \sin{\left(x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

-2*(3*sin(x) + (-1 + x)*cos(x))

$$- 2 \left(\left(x - 1\right) \cos{\left(x \right)} + 3 \sin{\left(x \right)}\right)$$

Simplify