By evaluating the limit of the difference quotient \lim_{h \to 0} \frac{f(x+h) - f(x)}{h},… — JAMB UTME

By evaluating the limit of the difference quotient $\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ , determine the value of the derivative of the function $f(x) = 2x^2 - 4x + 5$ at the point where $x = 3$ .

Answer: 8

Answer

The value of the derivative of

f(x) = 2x^2 - 4x + 5

x = 3

is 8.

Applying first principles,

f(x+h) - f(x) = 4xh + 2h^2 - 4h

. Dividing by

h

gives

4x + 2h - 4

. Taking the limit as

h \to 0

produces

f'(x) = 4x - 4

. Substituting

x = 3

yields

4(3) - 4 = 8

Step-by-Step Solution

Substitute

(x+h)

into the function

f(x) = 2x^2 - 4x + 5

and expand

f(x+h) = 2(x^2 + 2xh + h^2) - 4x - 4h + 5 = 2x^2 + 4xh + 2h^2 - 4x - 4h + 5

Apply algebraic expansion to determine the value of the function at

x+h

Form the difference

f(x+h) - f(x)

f(x+h) - f(x) = 4xh + 2h^2 - 4h

Subtract the original function terms to leave only terms containing

h

Divide the difference by

h

\frac{f(x+h) - f(x)}{h} = 4x + 2h - 4

Simplify the difference quotient prior to taking the limit.

Compute the limit as

h \to 0

f'(x) = 4x - 4

Taking the limit yields the general derivative function

f'(x)

Evaluate

f'(x)

x = 3

f'(3) = 4(3) - 4 = 8

Substitute

x = 3

to find the numerical rate of change at the given point.

Key Concept

Differentiation from First Principles

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