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 $f(x) = \frac{3}{x}$ for $x \neq 0$ ?

-\frac{3}{x^2}

Using the first principles formula

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

, substituting

f(x) = \frac{3}{x}

gives

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

Set up the first principles difference quotient

f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{h \to 0} \frac{\frac{3}{x+h} - \frac{3}{x}}{h}

Definition of differentiation from first principles.

Combine the fractions in the numerator using a common denominator

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

Algebraic fraction subtraction.

Divide by

h

and cancel the common factor

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

Simplifying the difference quotient before taking the limit.

Evaluate the limit as

h \to 0

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

Direct substitution of

h=0

after canceling

h

Differentiation of a reciprocal function using the first principles limit definition.

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