🔍 Unit 2: Derivatives

Evaluate \(\displaystyle \lim_{h\to 0}\frac{(2+h)^3-8}{h}\). What does this limit represent?

Rewrite each as a power of \(x\): a) \(\sqrt{x}\); b) \(\frac{1}{x^3}\); c) \(\sqrt[3]{x^5}\). Then state the derivative using the power rule.

Expand and simplify \((x+h)^3-x^3\). Then divide by \(h\) and simplify.

If \(f(x)=x^4\), use first principles to derive \(f'(x)\). Show all algebra and the limit step.

Without computing, predict whether the product rule or quotient rule should be used for: a) \((x^2+1)\sqrt{x}\); b) \(\frac{x+3}{x^2-1}\). Justify briefly.

In your own words, state the chain rule using both Leibniz \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\) and prime notation \((f\circ g)'(x)=f'(g(x))\,g'(x)\). Provide a simple example.

Which derivative rule do you find most confusing right now? What specifically is unclear?