If y = \frac{4x + 1}{x - 2}, calculate the numerical value of \frac{dy}{dx} at x = 3. — JAMB UTME

If $y = \frac{4x + 1}{x - 2}$ , calculate the numerical value of $\frac{dy}{dx}$ at $x = 3$ .

Answer: -9

The numerical value of

\frac{dy}{dx}

x = 3

-9

Using the quotient rule

\frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}

for

y = \frac{4x + 1}{x - 2}

gives

\frac{dy}{dx} = \frac{(x - 2)(4) - (4x + 1)(1)}{(x - 2)^2} = \frac{-9}{(x - 2)^2}

. Substituting

x = 3

produces

\frac{-9}{(3 - 2)^2} = -9

Identify

u(x)

and

v(x)

for the quotient rule formula.

u = 4x + 1

and

v = x - 2

The given equation

y = \frac{4x + 1}{x - 2}

is a quotient of two functions of

x

Calculate the individual derivatives

\frac{du}{dx}

and

\frac{dv}{dx}

\frac{du}{dx} = 4

and

\frac{dv}{dx} = 1

These derivatives are required components of the quotient rule.

Substitute the expressions into the quotient rule formula

\frac{dy}{dx} = \frac{v u' - u v'}{v^2}

and simplify.

\frac{dy}{dx} = \frac{(x - 2)(4) - (4x + 1)(1)}{(x - 2)^2} = \frac{4x - 8 - 4x - 1}{(x - 2)^2} = \frac{-9}{(x - 2)^2}

Simplifying the numerator yields the general derivative function.

Evaluate the derivative at

x = 3

\frac{dy}{dx}\Big|_{x=3} = \frac{-9}{(3 - 2)^2} = \frac{-9}{1} = -9

Substituting

x = 3

gives the requested numerical value.

Quotient Rule of Differentiation

Question