What is the derivative of the function f(x) = 3x^2 + 5x with respect to x, obtained using … — JAMB UTME

What is the derivative of the function $f(x) = 3x^2 + 5x$ with respect to $x$ , obtained using differentiation from first principles?

The derivative of the function is

6x + 5

Using the first-principles formula

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

, we expand

f(x+h) = 3(x+h)^2 + 5(x+h) = 3x^2 + 6xh + 3h^2 + 5x + 5h

. Subtracting

f(x) = 3x^2 + 5x

yields

6xh + 3h^2 + 5h

. Dividing by

h

produces

6x + 3h + 5

. Taking the limit as

h \to 0

gives

6x + 5

Evaluate

f(x+h)

for

f(x) = 3x^2 + 5x

f(x+h) = 3(x+h)^2 + 5(x+h) = 3(x^2 + 2xh + h^2) + 5x + 5h = 3x^2 + 6xh + 3h^2 + 5x + 5h

Substitute

(x+h)

into the original function expression and expand algebraically.

Find the difference

f(x+h) - f(x)

f(x+h) - f(x) = (3x^2 + 6xh + 3h^2 + 5x + 5h) - (3x^2 + 5x) = 6xh + 3h^2 + 5h

Subtract the original function

f(x)

to find the net change in output.

Divide the difference by

h

to set up the difference quotient

\frac{f(x+h) - f(x)}{h} = \frac{6xh + 3h^2 + 5h}{h} = 6x + 3h + 5

Cancel out

h

from each term in the numerator.

Take the limit as

h \to 0

f'(x) = \lim_{h \to 0} (6x + 3h + 5) = 6x + 5

Evaluate the expression as

h

approaches 0 to determine the instantaneous rate of change.

Differentiation from First Principles

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