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NSWMaths AdvancedQuick questions
Year 12: Calculus
Quick questions on Logarithmic and exponential calculus: derivatives and integrals of e^x and ln(x)
15short 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 standard derivatives?Show answer
$$\frac{d}{dx}(\ln x) = \frac{1}{x} \quad (x > 0)$$
What is composed derivatives (chain rule)?Show answer
$$\frac{d}{dx}(e^{f(x)}) = f'(x) e^{f(x)}$$
What is standard integrals?Show answer
$$\int \frac{1}{x} \, dx = \ln |x| + C$$
What is composed integrals?Show answer
$$\int e^{k x} \, dx = \frac{e^{k x}}{k} + C$$
What is non-$e$ bases?Show answer
For $a^x$ with $a > 0$, write $a^x = e^{x \ln a}$. Then
What is chain rule with $e$?Show answer
Differentiate $y = e^{\sin x}$.
What is product rule with $\ln$?Show answer
$u = x$, $v = \ln x$. $u' = 1$, $v' = \frac{1}{x}$.
What is quotient with $e$?Show answer
Differentiate $y = \frac{e^x}{x}$.
What is reverse chain rule integral?Show answer
Evaluate $\int \frac{3 x^2}{x^3 + 4} \, dx$.
What is definite integral?Show answer
Evaluate $\int_1^e \frac{1}{x} \, dx$.
What is $\int \frac{1}{x} \, dx \neq \frac{x^0}{0}$?Show answer
The power rule for integration excludes $n = -1$. The antiderivative of $\frac{1}{x}$ is $\ln |x|$, not a power.
What is forgetting absolute value bars?Show answer
Write $\int \frac{1}{x} \, dx = \ln |x| + C$. The bars are essential for intervals where $x < 0$.
What is treating $e^{2 x}$ like $2 e^x$?Show answer
They are different functions. $\frac{d}{dx}(e^{2 x}) = 2 e^{2 x}$ by the chain rule.
What is dropping the chain rule on $\ln$?Show answer
$\frac{d}{dx}(\ln(3 x)) = \frac{3}{3 x} = \frac{1}{x}$, but $\frac{d}{dx}(\ln(x^2)) = \frac{2 x}{x^2} = \frac{2}{x}$ (which also equals $\frac{d}{dx}(2 \ln x)$).
What is splitting $\ln $?Show answer
$\ln(a + b)$ does not equal $\ln a + \ln b$. Only $\ln(a b) = \ln a + \ln b$ and $\ln(a/b) = \ln a - \ln b$.