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 $f(x) = x^2 - 5x$ with respect to $x$ ?

The derivative of

f(x) = x^2 - 5x

with respect to

x

2x - 5

Evaluating the difference quotient

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

for

f(x) = x^2 - 5x

gives

\frac{x^2+2xh+h^2-5x-5h-(x^2-5x)}{h} = 2x + h - 5

. Taking the limit as

h \to 0

yields

2x - 5

Write down the definition of the derivative from first principles.

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

This is the standard formula for differentiation from first principles.

Substitute

f(x) = x^2 - 5x

and

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

into the definition.

\frac{f(x+h) - f(x)}{h} = \frac{[(x+h)^2 - 5(x+h)] - (x^2 - 5x)}{h}

This sets up the difference quotient for the given polynomial function.

Expand the numerator and combine like terms.

\frac{x^2 + 2xh + h^2 - 5x - 5h - x^2 + 5x}{h} = \frac{2xh + h^2 - 5h}{h} = 2x + h - 5

Expanding allows

x^2

and

-5x

terms to cancel out, leaving terms containing

h

Evaluate the limit as

h \to 0

\lim_{h \to 0} (2x + h - 5) = 2x - 5

h

approaches 0, the term

h

vanishes, giving the final derivative.

Differentiation from First Principles

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