1. Write the rules for differentiation for linear, power, exponential and logarithmic functions.
Power Rule: \(f(x)=x^a\Longrightarrow f'(x)=ax^{a-1}\), where \(a\) is constant
Linear: \(y=A+f(x) \Longrightarrow y'=f'(x)\); (constant multiples) \(y=Af(x) \Longrightarrow y'=Af'(x)\); F′(x)=f′(x)+g′(x)
Exponential: \(y=e^{g(x)} \Longrightarrow y^{'}=e^{g(x)}\cdot g^{'}(x)\) ; (general exponential function) \(f\left(x\right)=\ a^x\) --> \(f^{'}(x)=a^{x}\ln{a}\)
Logarithmic: \(g(x)=\ln{x} \Longrightarrow g^{'}(x)=\frac{1}{x}\)
2. 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: \(F(x)=\frac{f(x)}{g(x)}\Longrightarrow F'(x)=\frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}\)
3. Write the proof of the product rule
\(F'(x)=\lim_{h\to 0}\frac{F(x+h)-F(x)}{h}\Longrightarrow \lim_{h\to 0}\frac{f(x+h)\cdot g(x+h)-f(x)g(x)}{h}\)
\(F'(x)=\lim_{h\to 0}\frac{f(x+h)\cdot g(x+h)-f(x)g(x+h)+f(x)g(x+h)-f(x)g(x)}{h}\Longrightarrow\lim_{h\to 0}\frac{[f(x+h)-f(x)]g(x+h)+f(x)[g(x+h)-g(x)]}{h}\)
\(F'(x)=\lim_{h\to 0}g(x+h)\frac{f(x+h)-f(x)}{h}+f(x)\frac{g(x+h)-g(x)}{h}\)
\(F'(x)=g(x)f'(x)+f(x)g'(x)\)
4. Write the formula for the chain rule.
\(\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx\ }\ or\ y=F'\left(x\right)=f'\left(g\left(x\right)\right)\cdot g'\left(x\right)\)
5. Write the proof of the chain rule.