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The derivative is... \frac{2}{x} + 3x^2 + \frac{x}{x^2+1}

Yes, split with log laws... and more than you have suggested. But, imagine trying to do it without using log laws first, so you need to differentiate \frac{x^2e^{x^3}}{\sqrt{x^2+1}}...

\begin{align*} y &= x^k + (c - x)^k \\ \frac{dy}{dx} &= kx^{k - 1} + k(c - x)^{k - 1} \times -1 = k\left[x^{k - 1} - (c - x)^{k - 1}\right] \\ \\ \text{Put $\frac{dy}{dx} = 0$:} \quad k\left[x^{k - 1} - (c - x)^{k - 1}\right] &= 0 \\ \left(\frac{x}{c - x}\right)^{k - 1} &= 1 \qquad \text{as $k...

There is the evil differentiation question from Ascham (evil if you don't know what to do to work it out, I mean)... \text{Find } \frac{d}{dx}\left[\log_e{\left(\frac{x^2e^{x^3}}{\sqrt{x^2+1}}\right)}\right]

You've posted two question 10s... which one are you working on?

If I am selling 1 kg of cooking salt, I can label it and advertise with the claim "100% free of fat, sugar, and cholesterol" because that is true... even though no salt product for sale contains fat, sugar, or cholesterol.

Advertising is meant to be truthful, though it's often misleading or not entirely truthful. Ads can be taken down if proven to be false or misleading, but that requires a regulator to take action, so there is little prospect of action against carefully worded claims that don't make clear that...

Fees are based on demand and not on costs. Otherwise, law would be cheap.

You most certainly will notice it... Paying upfront (even part of it) used to attract a discount, I don't know if it still does. You will save money by paying upfront, but it is a lot of money to find if you are a student working to support yourself and living in one of the most expensive...

Sounds possible. Each Uni has its own system. USyd, for example, has a BSc(Hons) over 4 years with 192 credit points to graduate (48 credit points per year). Not sure if this still applies, but first year subjects were 6 points per semester, so 4 subjects for a full-time load each semester...

There is no teacher anywhere who suits every student that might see them. It's a matter not only of knowledge, but style, personality, engagement, etc, etc. As for tutoring places that have entrance exams and specialise in the most able students, there is no way to measure how those students...

Matching your response to the number of marks available is another important skill to develop. :)

Yes, I'm not surprised that only a brief answer was required, worth 1 to 2 marks. I just thought it was an interesting question to highlight some more of what is going on. :)

As jimmy has noted, the fees you pay are based on the subjects / courses / units of study you take. Since double degrees allow you to claim some of the subjects as credit towards both degrees, the total units completed is less than for each degree separately, and thus the total fees is also...

Bear in mind that the question asks for Arg(w1 - w2), with a capital A - not just for arg(w1 - w2). Thus, you seek not just the argument of the complex number, but specifically its principal argument. Now, as \alpha_1 and \alpha_2 are acute, \alpha_1 + \alpha_2 \in (0,\ \pi) \quad \implies...

That's a good question, though not one that really fits inside the HSC syllabus. Ring strain is a destabilising factor for ring systems except when they have six atoms in the ring - Wikipedia has an article about it - and the problem with bond angles away from the ideal tetrahedral angle is...

There are some schools that allow each student to come up with their own topic, so long as they can describe experiment(s) to answer the question they are posing and it is practicable to do. This makes for a LOT more work for the teachers, etc, and so many schools are greatly constraining the...

Or, a longer way... \begin{align*} \frac{z_2}{z_1} &= \frac{z_3}{z_2} \\ \arg{\left(\frac{z_2}{z_1}\right)} &= \arg{\left(\frac{z_3}{z_2}\right)} \\ \arg{z_2} - \arg{z_1} &= \arg{z_3} - \arg{z_2} \\ \arg{z_2} &= \frac{1}{2}\left(\arg{z_1} + \arg{z_3}\right) \\ \\ \text{Thus,} \quad...

Care to share any of the difficult questions?

There are hard things in Module 1 to test but many of them can't really be asked from only module 1 content