Using differentiation from first principles, what is the derivative of the function f(x) =… — JAMB UTME

Using differentiation from first principles, what is the derivative of the function $f(x) = x^2 + 3x$ with respect to $x$ ?

2x + 3

Applying the first principles limit formula

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

gives

\frac{x^2 + 2xh + h^2 + 3x + 3h - (x^2 + 3x)}{h} = \frac{2xh + h^2 + 3h}{h} = 2x + h + 3

. As

h \to 0

, this expression evaluates to

2x + 3

Set up the difference quotient using the definition of differentiation from first principles

\frac{f(x+h) - f(x)}{h} = \frac{[(x+h)^2 + 3(x+h)] - [x^2 + 3x]}{h}

The definition of derivative from first principles requires finding

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

Expand algebraic terms in the numerator

\frac{x^2 + 2xh + h^2 + 3x + 3h - x^2 - 3x}{h}

Expanding

(x+h)^2 = x^2 + 2xh + h^2

and

3(x+h) = 3x + 3h

allows terms to be combined.

Simplify the numerator by canceling like terms and dividing by

h

\frac{2xh + h^2 + 3h}{h} = 2x + h + 3

The terms

x^2 - x^2 = 0

and

3x - 3x = 0

cancel out, leaving terms containing

h

, which can be divided by

h

Evaluate the limit as

h \to 0

\lim_{h \to 0} (2x + h + 3) = 2x + 3

Setting

h = 0

in the simplified quotient yields the final derivative

f'(x) = 2x + 3

Differentiation from first principles

Question