- \(g(x)\) is a convex function over the interval I \( \Longleftrightarrow \) \(g^{''}(x)\ge 0,\ \forall x \in I\)
- We can categorize function over two dimensions: convexity vs concavity & increasing vs decreasing
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
- Why is \(e\) called the natural number?
- Take the exponential function \(e^x\)
- Let us now use Newton's quotient : \(\lim_{h\to 0}\frac{e^{x+h}-e^{x}}{h}\)
- We can rewrite this as \(\lim_{h\to 0}e^x\frac{e^{h}-1}{h}\) (Why? remember the properties of exponents)
- Taking the limit, we get \(f^{'}(x)=e^x\) (\(\lim_{h\to 0}\frac{e^h-1}{h}=1\))
- Thus, the exponential function is its own first derivative
Exponential functions and the chain rule
- Combining the rule for the differentiation of an exponential function and the chain rule, we can find the derivative for a function of the following form: \(y=e^{g(x)} \Longrightarrow y^{'}=e^{g(x)}\cdot g^{'}(x)\)
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
- We can generalize the discussion above to the function of the following form: \(f(x)=a^x\)
- \(f^{'}(x)=a^{x}\ln{a}\)
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
- \(g(x)=\ln{x} \Longrightarrow g^{'}(x)=\frac{1}{x}\)
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 \(y\):\(y^{'}=\left[\alpha\frac{A^{'}(x)}{A(x)}+\beta\frac{B^{'}(x)}{B(x)}+\gamma\frac{C^{'}(x)}{C(x)}\right]\cdot [A(x)]^{\alpha}[B(x)]^{\beta}[C(x)]^{\gamma}\)