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VICMath MethodsQuick questions
Unit 3
Quick questions on Differentiation from first principles: VCE Math Methods Unit 3
9short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.
What is worked example?Show answer
Differentiate $f(x) = x^2 + 3x$ from first principles.
What is step 1?Show answer
$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$.
What is step 2?Show answer
$f(x + h) = (x + h)^2 + 3(x + h) = x^2 + 2xh + h^2 + 3x + 3h$.
What is step 3?Show answer
$f(x + h) - f(x) = (x^2 + 2xh + h^2 + 3x + 3h) - (x^2 + 3x) = 2xh + h^2 + 3h = h(2x + h + 3)$.
What is step 4?Show answer
$f'(x) = \lim_{h \to 0} (2x + h + 3) = 2x + 3$.
What is skipping the limit definition?Show answer
"First principles" specifically means the limit. Writing $\frac{d}{dx}(x^2 + 3x) = 2x + 3$ by the power rule earns zero marks even if the answer is correct.
What is substituting $h = 0$ too early?Show answer
You cannot evaluate the difference quotient at $h = 0$ directly (you get $0/0$). Simplify first, then take the limit.
What is not factoring out $h$ before cancelling?Show answer
The whole point of the algebra is to cancel the $h$ in the denominator with one $h$ in the numerator. If you cannot factor $h$ out, recheck the algebra.
What is forgetting to expand $ ^n$ correctly?Show answer
Use the binomial expansion or pure algebra; do not write $(x + h)^2 = x^2 + h^2$.