Mister Exam

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The derivative of the function/ 2*x*cos(x)

Derivative of 2xcos(x)

The solution

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2*x*cos(x)

$$2 x \cos{\left(x \right)}$$

(2*x)*cos(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = 2 x$ ; 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. Apply the power rule: $x$ goes to $1$
  So, the result is: $2$
$g{\left(x \right)} = \cos{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of cosine is negative sine:
  
  $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
The result is: $- 2 x \sin{\left(x \right)} + 2 \cos{\left(x \right)}$

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Derivative of 2*x*cos(x)