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

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

Answer: 8

The derivative of

f(x) = x^2 + 2x

evaluated at

x = 3

is 8.

Applying first principles to

f(x) = x^2 + 2x

yields

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

. Evaluating this derivative at

x = 3

gives

2(3) + 2 = 8

Apply the definition of differentiation from first principles

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

First principles uses the limit of the difference quotient to compute the instantaneous rate of change.

Substitute

f(x) = x^2 + 2x

into the difference quotient

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

Expanding terms allows cancellation of non-

h

terms in the numerator.

Simplify the fraction by dividing by

h

2x + h + 2

Dividing out

h

removes the indeterminate form

\frac{0}{0}

Evaluate the limit as

h \to 0

and substitute

x = 3

f'(3) = 2(3) + 2 = 8

Setting

h = 0

yields the derivative function

f'(x) = 2x + 2

, which evaluates to 8 at

x = 3

Differentiation from First Principles

Question