Using differentiation from first principles, evaluate the value of the derivative of the f… — JAMB UTME

Using differentiation from first principles, evaluate the value of the derivative of the function $f(x) = 2x^2 + 3x - 1$ at the point where $x = 1$ .

Cevap: 7

The value of the derivative at

x = 1

is 7.

Using first principles, the derivative is defined as

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

. Evaluating at

x = 1

, we find

f(1) = 4

and

f(1+h) = 4 + 7h + 2h^2

. Subtracting

f(1)

leaves

7h + 2h^2

, and dividing by

h

yields

7 + 2h

. Taking the limit as

h \to 0

yields the final result of 7.

Calculate the value of the function at

x = 1

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

This establishes the base value needed for the difference quotient.

Expand and simplify

f(1+h)

f(1+h) = 2(1+h)^2 + 3(1+h) - 1 = 4 + 7h + 2h^2

This gives the value of the function at the incremented point

x + h

Form and simplify the difference quotient

\frac{f(1+h) - f(1)}{h}

\frac{4 + 7h + 2h^2 - 4}{h} = 7 + 2h

Dividing by

h

eliminates the indeterminate form before taking the limit.

Evaluate the limit as

h \to 0

f'(1) = \lim_{h \to 0} (7 + 2h) = 7

Taking

h = 0

in the simplified quotient gives the exact rate of change at

x = 1

Differentiation from First Principles

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