Question

F’(x)=0,5x умножить на cos2x

Answer 1

F'(x)=0.5*cos2x-0.5x*2*sin2x=0.5cos2x-x*sin2x

Answer 2

To find the derivative of $F(x) = 0.5x \cdot \cos(2x)$, we'll use the product rule and the chain rule. The product rule states that if you have two functions of $x$, $u(x)$ and $v(x)$, then the derivative of their product is given by:

$\frac{d}{dx}(u(x) \cdot v(x)) = u'(x) \cdot v(x) + u(x) \cdot v'(x)$

In this case, $u(x) = 0.5x$ and $v(x) = \cos(2x)$. Let's find the derivatives of $u(x)$ and $v(x)$ first:

$u'(x) = 0.5$ $v'(x) = -2\sin(2x)$ (using the chain rule)

Now, we can apply the product rule:

$F'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)$ $F'(x) = 0.5 \cdot \cos(2x) + 0.5x \cdot (-2\sin(2x))$ $F'(x) = 0.5 \cdot \cos(2x) - x \cdot \sin(2x)$