Derivative of 5sinx*(2-3x)

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5*sin(x)*(2 - 3*x)

$$\left(2 - 3 x\right) 5 \sin{\left(x \right)}$$

(5*sin(x))*(2 - 3*x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = 5 \sin{\left(x \right)}$ ; 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. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  So, the result is: $5 \cos{\left(x \right)}$
$g{\left(x \right)} = 2 - 3 x$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $2 - 3 x$ term by term:
  1. The derivative of the constant $2$ is zero.
  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: $-3$
The result is: $5 \left(2 - 3 x\right) \cos{\left(x \right)} - 15 \sin{\left(x \right)}$
Now simplify:

$- 5 \left(3 x - 2\right) \cos{\left(x \right)} - 15 \sin{\left(x \right)}$

The answer is:

$- 5 \left(3 x - 2\right) \cos{\left(x \right)} - 15 \sin{\left(x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

-15*sin(x) + 5*(2 - 3*x)*cos(x)

$$5 \left(2 - 3 x\right) \cos{\left(x \right)} - 15 \sin{\left(x \right)}$$

Simplify

The second derivative [src]

5*(-6*cos(x) + (-2 + 3*x)*sin(x))

$$5 \left(\left(3 x - 2\right) \sin{\left(x \right)} - 6 \cos{\left(x \right)}\right)$$

Simplify

The third derivative [src]

5*(9*sin(x) + (-2 + 3*x)*cos(x))

$$5 \left(\left(3 x - 2\right) \cos{\left(x \right)} + 9 \sin{\left(x \right)}\right)$$

Simplify