If y = (2x - 3)^5, what is the value of \frac{dy}{dx} at x = 2? — JAMB UTME

If $y = (2x - 3)^5$ , what is the value of $\frac{dy}{dx}$ at $x = 2$ ?

The derivative of

y = (2x - 3)^5

with respect to

x

requires the chain rule:

\frac{dy}{dx} = 5(2x - 3)^4 \cdot \frac{d}{dx}(2x - 3) = 5(2x - 3)^4 \cdot 2 = 10(2x - 3)^4

. Substituting

x = 2

gives

10(2(2) - 3)^4 = 10(1)^4 = 10

, which makes

10

the correct value.

Identify the inner function

u(x)

and outer function

f(u)

Let

u = 2x - 3

, so

y = u^5

The chain rule states that

\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

Differentiate

y

with respect to

u

and

u

with respect to

x

\frac{dy}{du} = 5u^4 = 5(2x - 3)^4

and

\frac{du}{dx} = 2

Apply the power rule to both functions.

Multiply the derivatives to find

\frac{dy}{dx}

\frac{dy}{dx} = 5(2x - 3)^4 \cdot 2 = 10(2x - 3)^4

Combine terms using the chain rule formula.

Substitute

x = 2

into the derivative

\frac{dy}{dx}\Big|_{x=2} = 10(2(2) - 3)^4 = 10(1)^4 = 10

Evaluate the expression numerically at the target point.

Chain Rule for Differentiation

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