Using differentiation from first principles, what is the numerical value of the derivative… — JAMB UTME

Using differentiation from first principles, what is the numerical value of the derivative of the function $f(x) = 3x^2 - 4x + 1$ at $x = 2$ ?

Answer: 8

The numerical value of the derivative of

f(x) = 3x^2 - 4x + 1

x = 2

is 8.

Using the first-principles limit definition, the increment

f(2+h) - f(2)

simplifies to

8h + 3h^2

. Dividing by

h

yields

8 + 3h

, which evaluates to 8 as

h \to 0

Calculate

f(2)

f(2) = 5

Substitute

x = 2

into

f(x) = 3x^2 - 4x + 1

Expand

f(2+h)

f(2+h) = 5 + 8h + 3h^2

Substitute

x = 2+h

into

f(x)

and expand algebraically.

Simplify the difference quotient

\frac{f(2+h) - f(2)}{h}

\frac{8h + 3h^2}{h} = 8 + 3h

Subtract

f(2)

from

f(2+h)

and divide every term by

h

Evaluate the limit as

h \to 0

f'(2) = 8

h

approaches 0, the term

3h

vanishes, leaving 8.

Differentiation from First Principles

Estimated Time:1m 30s

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