1.
Differentiation of a...
Linear function:  \(y=ax+b\ ->\ y'=a\)
Power function: \(f\left(x\right)=x^a\ ->\ f'\left(x\right)=ax^{a-1}\), where a  is a constant
Exponential function:\(f\left(x\right)=a^x->\ f'\left(x\right)\ =\ a^x\ln a\)
Logarithmic function:   \(g\left(x\right)=\ln x\ ->\ g'\left(x\right)=\ \frac{1}{x}\)
2. Product Rule: \(\left(f\cdot g\right)'=f'\cdot g+f\cdot g'\)
Quotient Rule: \(\left(\frac{f}{g}\right)'=\frac{\left(f'\cdot g+f\cdot g'\right)}{g^2}\)
3. Product Rule Proof
\(\frac{d}{dx}\left[f\left(x\right)\right]=\lim_{h->0}\left(\frac{\left(f\left(x+h\right)-f\left(x\right)\right)}{h}\right)\)
\(\frac{d}{dx}\left[f\left(x\right)g\left(x\right)\right]=\) \(\lim_{h->0}\)\(\frac{\left(f\left(x+h\right)g\left(x+h\right)-f\left(x+h\right)g\left(x\right)+f\left(x+h\right)g\left(x\right)-f\left(x\right)g\left(x\right)\right)}{h}\)
\(\lim_{h->0}\)\(\left(f\left(x+h\right)\frac{g\left(x+h\right)-g\left(x\right)}{h}\right)+\left(g\left(x\right)\frac{\left(f\left(x+h\right)-f\left(x\right)\right)}{h}\right)\)
\(\left(\lim_{h->0}f\left(x+h\right)\right)\cdot\left(\lim_{h->0}\frac{\left(g\left(x+h\right)-g\left(x\right)\right)}{h}\right)\)+\(\left(\lim_{x->0}g\left(x\right)\right)\cdot\left(\lim_{h->0}\frac{\left(f\left(x+h\right)-f\left(x\right)\right)}{h}\right)\)
\(f\left(x\right)g'\left(x\right)+g\left(x\right)f'\left(x\right)\)   
4. Chain rule = \(f\left(g\left(x\right)\right)\ =\ f'\left(g\left(x\right)\right)\cdot g'\left(x\right)\)
5. Chain rule proof
\(\frac{d}{dx}\left[y\left(u\left(x\right)\right)\right]=\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\)
\(\frac{dy}{dx}=\lim_{\ \ x->0}\left(\frac{\ \ y}{\ \ x}\right)\)
\(Lim_{\ \ x->0}\left(\frac{\ \ y}{\ \ u}\right)\cdot\left(\frac{\ \ u}{\ \ x}\right)\)
\(\left(\lim_{\ \ x->0}\left(\frac{\ \ y}{\ \ u}\right)\right)\cdot\left(\lim_{\ \ x->0}\left(\frac{\ \ u}{\ \ x}\right)\right)\)
\(\frac{dy}{du}\cdot\frac{du}{dx}\)
6.
Price elasticity of demand: \(\frac{\ \ D}{\ \ P}\cdot\frac{P}{D}\)
Income elasticity of demand: \(\frac{\ \ P}{\ \ M}\cdot\frac{M}{D}\)
Cross price elasticity of demand:  \(E_{A,B}=\frac{\frac{\left(Q_f-Q_i\right)}{\left(Q_f+Q_i\right)\cdot\frac{1}{2}}}{\frac{\left(P_f-P_i\right)}{\left(P_f+P_i\cdot\frac{1}{2}\right)}}\)
 
7.
The Polonius point is where the consumer neither borrows nor lends because he simply consumes all the income he has.
\(u\left(c1,c2\right)=c_1^{\alpha}c_2^{\beta}\)
\(\ln\left(u\right)=\ln c_1+\frac{1}{1+p}\ln c_2\)
\(y_1+\frac{y_2}{1+r}=c_1+\frac{c_2}{1+r}\)
\(c_2=y_2-\left(1+r\right)\left(c_1-y_1\right)\)
 \(u\left(c_1,c_2\right)=\ln c_1+\frac{1}{1+p}\left(\ln\left(y_2-\left(1+r\right)\left(c1-y1\right)\right)\right)\)
FOC
\(\frac{\ \ u}{\ \ c_1}=\frac{1}{c_1}+\frac{1}{1+p}\cdot\frac{1}{y_2-\left(1+r\right)\left(c_1-y_1\right)}\cdot\left(-\left(1+r\right)\right)=0\)
\(c_1=\frac{\left(y_2+y_1\right)}{2+p}\)
\(\frac{c_2}{c_1}=\frac{\left(1+r\right)}{\left(1+p\right)}\)
\(\frac{\left(change\ in\ C\ +c_1\right)}{c_1}=\frac{\left(1+r\right)}{\left(1+p\right)}\)
\(\frac{\left(delta\ C\right)}{C_1}+1=\frac{\left(1+r\right)}{\left(1+p\right)}\)
\(\frac{\left(delta\ C\right)}{c_1}=\frac{\left(1+r\right)}{\left(1+p\right)}-1\)
8.  If \(\lim_{x->\ c}f\left(x\right)=0\) and \(\lim_{x->c}g\left(x\right)\ =0\) and \(\lim_{x->c}\left(\frac{f'\left(x\right)}{g'\left(x\right)}\right)=L\) Then \(Lim_{x->c}\left(\frac{f\left(x\right)}{g\left(x\right)}\right)=L\)
And
If 
\(Lim_{x->0}\left(\frac{\left(e^x-1\right)}{x}\right)\)
\(f\left(x\right)=e^x-1\)
\(f'\left(x\right)=e^x\)
g(x) =x
\(g'\left(x\right)=1\)
so \(Lim_{x->0}\left(\frac{e^x}{1}\right)\ =1\)
Thus \(Lim_{x->0}\left(\frac{\left(e^x-1\right)}{x}\right)=1\)
9. 
Concave: a function if the line segment joining any two points on the graph is below the graph or on the graph thus \(f(x1)-f\left(x2\right)\le f'\left(x2\right)\left(x1-x2\right)\) for all \(x1\ \ I\ and\ x2\ \ I\) and if and only if \(f''\left(x\right)\le0\ \) for all x in the interior of \(I\).
Convex: a function if the line segment joining any two points on the graph is above the graph or on the graph thus \(f\left(x1\right)-f\left(x2\right)\ge f'\left(x2\right)\left(x1-x2\right)\) for all \(x1\ \ \ \ I\) and \(x2\ \ \ \ I\) and if and only if \(f''\left(x\right)\ge\ 0\) for all x in the interior of \(I\).
Strictly concave: If the segment joining any two points on the graph is below the graph thus  if for all a    I, all b    I with a    b, and all λ    (0,1) we have \(f((1−λ)a+λb)>(1−λ)f(a)+λf(b)\) and if and only if \(f''\left(x\right)\ <\ 0\) 
Strictly convex: If the segment joining any two points on the graph is above the graph thus  if for all a    I, all b    I with a    b, and all λ    (0,1) we have
\(f((1−λ)a+λb)<(1−λ)f(a)+λf(b)\) and if and only if \(f''\left(x\right)\ >\ 0\)