Write the formula for the product rule and the quotient rule of differentiation.
Product Rule Formula: \(F(x)=f(x)\cdot g(x)\Longrightarrow F'(x)=f'(x)g(x)+f(x)g'(x)\)
Quotient Rule Formula:   \(F(x)=\frac{f(x)}{g(x)}\Longrightarrow F'(x)=\frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}\)
 
  1.  Write the proof of the product rule
\(F'(x)=\lim_{h\to 0}\frac{F(x+h)-F(x)}{h}\Longrightarrow \lim_{h\to 0}\frac{f(x+h)\cdot g(x+h)-f(x)g(x)}{h}\)
\(F'(x)=\lim_{h\to 0}\frac{f(x+h)\cdot g(x+h)-f(x)g(x+h)+f(x)g(x+h)-f(x)g(x)}{h}\Longrightarrow\lim_{h\to 0}\frac{[f(x+h)-f(x)]g(x+h)+f(x)[g(x+h)-g(x)]}{h}\)
\(F'(x)=\lim_{h\to 0}g(x+h)\frac{f(x+h)-f(x)}{h}+f(x)\frac{g(x+h)-g(x)}{h}\)
\(F'(x)=g(x)f'(x)+f(x)g'(x)\)
Write the formula for the chain rule.
\(\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx\ }\ or\ y=F\left(x\right)=f\left(g\left(x\right)\right)\)
Write the proof of the chain rule.