Examples

Establish whether the following function are concave, convex, decreasing, or decreasing
a) \(f(x)=x^2-2x+2\)
\(f^{'}(x)=2x-2\)
\(f^{''}(x)=2\)
b) \(f(x)=ax^2+bx+c\)
\(f^{'}(x)=2ax+b\)
\(f^{''}(x)=2a\)
c) \(Y=AK^{\alpha}\)
\(Y^{'}=\alpha AK^{\alpha-1}\)
\(Y^{''}=\alpha(\alpha-1)AK^{\alpha-2}\)
d) Suppose the function U and g are both increasing and concave, with \(U^{'}\ge 0,\ U^{''}\leq 0,\ g^{'}\ge 0,\ g^{''}\leq 0\). Prove that the composite function \(f(x)=g(U(x))\) is also increasing and concave.
e) Take the fourth order derivative of the following function:
\(f(x)=3x^{-1}+6x^{3}-x^2\)

Exponential functions

Exponential functions and the chain rule

Examples

Take the derivative of the following functions
a) \(y=e^{-x}\)
b) \(y=x^p e^{ax}\)
c) \(y=\sqrt{e^{2x}+x}\)

General exponential functions

Examples

Take the derivative of the following functions
a) \(f(x)=10^{-x}\)
\(f^{'}(x)=10^{-x}\ln{10}\cdot (-1)=-\ln{10}\cdot 10^{-x}\) (remember to use the chain rule!)
b) \(g(x)=x2^{3x}\)
\(g^{'}(x)=2^{3x}+x(2^{3x}\ln{2}\cdot 3)\)

Logarithmic functions

Examples

Compute the first and second derivatives of the following functions
a) \(y=x^3+\ln x\)
b) \(y=x^2\ln x\)
c) \(y=\frac{\ln x}{x}\)
d) \(y=[A(x)]^{\alpha}[B(x)]^{\beta}[C(x)]^{\gamma}\)
Take log of both sides: \(\ln y=\alpha \ln A(x)+ \beta \ln B(x)+ \gamma \ln C(x)\)
Use the rules for logarithmic functions
\(\frac{y^{'}}{y}=\alpha\frac{A^{'}(x)}{A(x)}+\beta \frac{B^{'}(x)}{B(x)}+\gamma \frac{C^{'}(x)}{C(x)}\)
Multiply both sides by \(y\):\(y^{'}=\left[\alpha\frac{A^{'}(x)}{A(x)}+\beta\frac{B^{'}(x)}{B(x)}+\gamma\frac{C^{'}(x)}{C(x)}\right]\cdot [A(x)]^{\alpha}[B(x)]^{\beta}[C(x)]^{\gamma}\)