Given the function y = \frac{2x - 3}{(x^2 + 1)^2}, find the numerical value of \frac{dy}{d… — JAMB UTME

Given the function $y = \frac{2x - 3}{(x^2 + 1)^2}$ , find the numerical value of $\frac{dy}{dx}$ at $x = 1$ .

Answer: 1

The numerical value of

\frac{dy}{dx}

x = 1

is 1.

Applying the quotient rule

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

alongside the chain rule for the denominator yields

\frac{dy}{dx} = \frac{2(x^2 + 1)^2 - (2x - 3) \cdot 4x(x^2 + 1)}{(x^2 + 1)^4}

. Substituting

x = 1

evaluates to

\frac{2(4) - (-1)(8)}{16} = \frac{16}{16} = 1

Identify the numerator and denominator functions

Let

u(x) = 2x - 3

and

v(x) = (x^2 + 1)^2

The given function is structured as a quotient

y = \frac{u}{v}

, requiring the quotient rule.

Find the derivatives

u'(x)

and

v'(x)

u'(x) = 2

and

v'(x) = 2(x^2 + 1) \cdot 2x = 4x(x^2 + 1)

Differentiating

u(x)

follows standard polynomial rules;

v(x)

requires the chain rule.

Evaluate

u(1), u'(1), v(1),

and

v'(1)

x = 1

u(1) = -1

u'(1) = 2

v(1) = 4

, and

v'(1) = 8

Evaluating components before substitution simplifies the arithmetic.

Apply the quotient rule formula

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

x = 1

\frac{dy}{dx} = \frac{(2)(4) - (-1)(8)}{4^2} = \frac{8 + 8}{16} = 1.

Substitute the calculated component values into the quotient rule formula.

Quotient Rule and Chain Rule of Differentiation

Estimated Time:1m 30s

Question