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)