Back to the full dot-point answer

NSWMaths Extension 1Quick questions

Calculus (ME-C1, C2, C3)

Quick questions on Derivatives and integrals of inverse trigonometric functions

13short 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
$ddx(arcsin⁡x)=11−x2,x∈(−1,1).\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1 - x^2}}, \qquad x \in (-1, 1).$
What is derivation (sketch)?
Show answer
Let y=arcsin⁡xy = \arcsin x, so sin⁡y=x\sin y = x. Differentiate implicitly:
What is chain rule extensions?
Show answer
$ddx(arcsin⁡f(x))=f′(x)1−[f(x)]2,\frac{d}{dx}\bigl(\arcsin f(x)\bigr) = \frac{f'(x)}{\sqrt{1 - [f(x)]^2}},$
What is standard antiderivatives?
Show answer
$∫11−x2 dx=arcsin⁡x+C,\int \frac{1}{\sqrt{1 - x^2}} \, dx = \arcsin x + C,$
What is chain rule on arctan⁡\arctan?
Show answer
ddx(arctan⁡(x2))=2x1+x4\frac{d}{dx}\bigl(\arctan(x^2)\bigr) = \frac{2 x}{1 + x^4}.
What is product rule combination?
Show answer
ddx(xarctan⁡x)=arctan⁡x+x⋅11+x2=arctan⁡x+x1+x2\frac{d}{dx}\bigl(x \arctan x\bigr) = \arctan x + x \cdot \frac{1}{1 + x^2} = \arctan x + \frac{x}{1 + x^2}.
What is implicit derivative?
Show answer
Find dydx\frac{dy}{dx} if y=arcsin⁡ ⁣(x2)y = \arcsin\!\left( \frac{x}{2} \right) at x=1x = 1.
What is integral?
Show answer
Evaluate ∫01/211−4x2 dx\int_0^{1/2} \frac{1}{\sqrt{1 - 4 x^2}} \, dx.
What is arctan integral?
Show answer
Evaluate ∫0314+x2 dx\int_0^{\sqrt{3}} \frac{1}{4 + x^2} \, dx.
What is domain ignored?
Show answer
arcsin⁡\arcsin derivative is undefined at x=±1x = \pm 1, where the tangent is vertical.
What is missing chain-rule factor?
Show answer
ddx(arcsin⁡(g(x)))\frac{d}{dx}(\arcsin(g(x))) includes g′(x)g'(x) in the numerator, not just 11−g2\frac{1}{\sqrt{1 - g^2}}.
What is wrong aa in the integral formula?
Show answer
∫dx4+x2\int \frac{dx}{4 + x^2} has a=2a = 2 (not 44), so the constant in front of arctan⁡\arctan is 12\frac{1}{2} (not 14\frac{1}{4}).
What is forgetting absolute value with related logs?
Show answer
When the integrand is f′(x)f(x)\frac{f'(x)}{f(x)}, the antiderivative is ln⁡∣f(x)∣\ln |f(x)|, not an inverse-trig form. :::

All Maths Extension 1Q&A pages