Power:
- \(f(x)=x^a\Longrightarrow f'(x)=ax^{a-1}\), wehere \(a\) is constant
Exponential:
Logarithmic:
2. Formula for the product rule and the quotient rule of differentiation
3. Proof for the product rule
4. Formula for the chain rule
\(\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}\) \(\)
\(or\)
\(y=F(x)=f(g(x))\)Thus, \(F^{'}(x)=f^{'}(g(x))\cdot g^{'}(x)\)
5. Proof for the chain rule
Write \(F(x)=f(g(x))\)
Use Newton's quotient to write \(F^{'}(x)=\lim_{h\to 0} \frac{f(g(x+h))-f(g(x))}{h}\)
Define \(k=g(x+h)-g(x)\) and substitute \(g(x+h)=g(x)+k\) into \(f(g(x+h))\)
Multiply and divide \(F^{'}(\cdot)\) by \(k\): \(F^{'}(x)=\lim_{h\to 0} \frac{f(g(x)+k)-f(g(x))}{k}\cdot \frac{k}{h}=F^{'}(x)=\lim_{h\to 0} \frac{f(g(x+h))-f(g(x))}{k}\cdot \frac{g(x+h)-g(x)}{h}\)
Taking the limit yields: \(F^{'}(\cdot)=f^{'}(g(x))\cdot g^{'}(x)\)
6. Formula for price elasticity of demand? What the one for the income elasticity of demand? What is the cross price elasticity of demand? Use: D:Q(P)
7. What is “Polonius’ point”? Set up the intertemporal utility maximization problem for an individual over two periods (i.e., t = 1, 2). Find the formula for consumption in the first period as a function of the interest rate, the discount rate, and income over the two periods.
8. What is l’Hopital’s Rule? Use the rule to find the following limit: lim \(\lim_{x_{\vec{ }}0}\frac{e^x-1}{x}\)
9. Define convex, concave, strictly convex, and strictly concave curves using the notion of derivative (hint: not just first derivative!)
\(f(x)\) is a concave function over the interval I \( \Longleftrightarrow \) \(f^{''}(x)\leq 0,\ \forall x \in I\)
\(\)
\(\)\(g(x)\) is a convex function over the interval I \( \Longleftrightarrow \) \(g''(x)\ge 0,\ \forall x \in I\)