Using differentiation from first principles, evaluate the value of the derivative \frac{dy… — JAMB UTME

Using differentiation from first principles, evaluate the value of the derivative $\frac{dy}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ for the cubic function $f(x) = 2x^3 - 9x^2 + 12x - 5$ at the point $x = 3$ .

Answer: 12

Answer

The derivative evaluated at

x = 3

is equal to 12.

Evaluating the definition of the derivative from first principles for

f(x) = 2x^3 - 9x^2 + 12x - 5

yields

\lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = 6x^2 - 18x + 12

. Substituting

x = 3

gives

6(3)^2 - 18(3) + 12 = 54 - 54 + 12 = 12

Step-by-Step Solution

Set up the difference quotient definition from first principles

f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

Differentiation from first principles requires finding the limit of the average rate of change as the increment

h

approaches zero.

Substitute

(x+h)

into

f(x) = 2x^3 - 9x^2 + 12x - 5

and expand

f(x+h) = 2x^3 + 6x^2h + 6xh^2 + 2h^3 - 9x^2 - 18xh - 9h^2 + 12x + 12h - 5

Expanding binomial terms

(x+h)^3

and

(x+h)^2

reveals all components involving

h

Calculate

f(x+h) - f(x)

and factor out

h

f(x+h) - f(x) = h(6x^2 + 6xh + 2h^2 - 18x - 9h + 12)

Terms independent of

h

cancel out completely, isolating

h

as a common factor.

Divide by

h

and evaluate the limit as

h \to 0

f'(x) = 6x^2 - 18x + 12

Canceling

h

resolves the

\frac{0}{0}

indeterminate form, allowing direct substitution of

h=0

Substitute

x = 3

into

f'(x)

f'(3) = 6(3)^2 - 18(3) + 12 = 12

Evaluating at

x = 3

gives the numerical value of the instantaneous rate of change at that specific point.

Key Concept

Differentiation from First Principles

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