Question

В точке x=2 вычислите значение производной функции f(x) = 4x+7/2x-1

Answer 1

Решение задания смотри на фотографии

Answer 2

To find the derivative of the given function $f(x) = \frac{4x+7}{2x-1}$, we'll use the quotient rule. The quotient rule states that for functions $u(x)$ and $v(x)$:

$\frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{v(x)\frac{du(x)}{dx} - u(x)\frac{dv(x)}{dx}}{v(x)^2}$

Let's differentiate $u(x) = 4x + 7$ and $v(x) = 2x - 1$:

$\frac{du(x)}{dx} = 4$ $\frac{dv(x)}{dx} = 2$

Now, plug these values into the quotient rule formula:

$\frac{d}{dx}\left(\frac{4x+7}{2x-1}\right) = \frac{(2x-1)(4) - (4x+7)(2)}{(2x-1)^2}$

Simplify the expression:

$\frac{d}{dx}\left(\frac{4x+7}{2x-1}\right) = \frac{8x - 4 - 8x - 14}{(2x-1)^2} = \frac{-18}{(2x-1)^2}$

Now, to find the value of the derivative at $x = 2$, plug $x = 2$ into the expression:

$f'(2) = \frac{-18}{(2(2)-1)^2} = \frac{-18}{(4-1)^2} = \frac{-18}{3^2} = \frac{-18}{9} = -2$

Therefore, the value of the derivative at $x = 2$ is $f'(2) = -2$.