Q.Differentiate $f(x) = 3x^2 + 2x$ from first principles.

Apply the limit definition of the derivative directly, expanding $f(x+h)$ and simplifying before letting $h\to0$.

$f'(x) = 6x+2$

Given: $f(x) = 3x^2+2x$.

Step 1 — Write $f(x+h)$: $f(x+h) = 3(x+h)^2+2(x+h) = 3x^2+6xh+3h^2+2x+2h$.

Step 2 — Subtract $f(x)=3x^2+2x$: $f(x+h)-f(x) = 6xh+3h^2+2h$.

Step 3 — Divide by $h$: $\dfrac{6xh+3h^2+2h}{h} = 6x+3h+2$.

Step 4 — Let $h\to0$: $f'(x)=6x+2$.

Check (independent method — power and sum rule): $\dfrac{d}{dx}(3x^2)=6x$ and $\dfrac{d}{dx}(2x)=2$, so $f'(x)=6x+2$ — matches the first-principles result exactly.

Dropping the $3h^2$ term before dividing by $h$ — every term in the expansion must be carried through the subtraction and division steps; only terms that STILL contain $h$ after dividing vanish in the final limit step.