Slopes of curves

Example

Tangents

Example

Derive the derivative of \(f(x)=x^2\) using the method above
\(f(x)=x^2\) implies \(f(a+h)=(a+h)^2\) and \(f(a)=a^2\).
Substituting in the formula for the derivative yields
\(f'(a)=\lim_{h\to 0}\frac{(a+h)^2-a^2}{h}\)
Basic arithmetic yields
\(\)\(f'(a)=\lim_{h\to 0}\frac{a^2+2ah+h^2-a^2}{h}=\lim_{h\to 0}\frac{2ah+h^2}{h}=\lim_{h\to 0}\frac{2a+h}{1}=2a\)
  1. Add \(h\neq 0\) to \(a\) and compute \(f(a+h)\)
  2. Compute \(f(a+h)-f(a)\)
  3. Form the Newton quotient
  4. Simplify the resulting formula as much as possible, making sure not to have \(h\) in the denominator
  5. Take the limit as h tends to 0

Example

Use these rules to find the derivative of \(f(x)=x^3\)

Some notational issues

Increasing and decreasing functions

We can reformulate this conditions in terms of the function's derivative:

Examples

  1. Find whether the function \(f(x)=\frac{1}{2}x^2-2\) is increasing or decreasing
  2. Find whether the function \(f(x)=-\frac{1}{3}x^3+2x^2-3x+1\) is increasing or decreasing

Rates of change

Rate of change interpretation

Example

Consider a firm producing some commodity in a given period, with
We can use Newton's quotient to find the marginal cost of the firm:
\(C'(x)=\lim_{h\to 0}\frac{C(x+h)-C(h)}{x}\)
For an \(h\) small enough, we can write
\(C'(x)\simeq \frac{C(x+h)-C(x)}{h}\)
For \(h=1\) we obtain
\(C'(x)\simeq C(x+1)-C(x)\)
Thus, the interpretation of marginal cost is (approximately) the increase in cost due to an increase in output by one unit

Differentiability and empirical functions

Economists use derivatives to study the rate of change of a wide variety of variables. In some case, this is mathematically improper, as the variable of interest is discrete and not continuous and we cannot define a derivative for discrete functions 

A dash of limits

\(\lim_{x\to a} f(x)=A\) or \(f(x) \to A\) as \(x\to a\)

Rules of limits

If \(\lim_{x\to a}f(x)=A\) and \(\lim_{x\to a}g(x)=B\), then
  1. \(\lim_{x\to a}(f(x)\pm g(x))=A\pm B\)
  2. \(\lim_{x\to a}(f(x)\cdot g(x))=A\cdot B\)
  3. \(\lim_{x\to a}\frac{f(x)}{g(x)}=A/B, B\neq 0\)
  4. \(\lim_{x\to a}(f(x))^r=A^r\)

Examples

  1. \(\lim_{x\to 2}(x^2+5x)\)
  2. \(\lim_{x\to 4}\frac{2x^{3/2}-\sqrt{x}}{x^2-15}\)
  3. \(\lim_{x\to a}Ax^n\)

Simple rules of differentiation

Some rules

\(y=A+f(x) \Longrightarrow y'=f'(x)\)
\(y=Af(x) \Longrightarrow y'=Af'(x)\)

Proof

\(f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}\)
Now write
\(g(a)=Af(a)\)
Which implies
\(g'(a)=\lim_{h\to 0}\frac{g(a+h)-g(a)}{h}\)
Substituting from above yields
\(g'(a)=\lim_{h\to 0}\frac{Af(a+h)-Af(a)}{h} \Longrightarrow g'(a)=\lim_{h\to 0}A\frac{f(a+h)-f(a)}{h}\) 
Or
\(g'(a)=A\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}=Af'(a)\)
QED

Examples

Find the derivative of the following functions
  1. \(y=5+f(x)\)
  2. \(y=f(x)-\frac{1}{2}\)
  3. \(y=4f(x)\)
  4. \(y=\frac{Af(x)+B}{C}\)

The power rule

\(f(x)=x^a\Longrightarrow f'(x)=ax^{a-1}\), wehere \(a\) is constant

Examples

Find the derivative of the following functions
  1. \(y=x^5\)
  2. \(y=3x^8\)
  3. \(y=\frac{x^{100}}{100}\)

Sums, products, and quotients

Differentiation of sums and differences

\(F(x)=f(x)\pm g(x) \Longrightarrow f'(x)\pm g'(x)\)

Proof

\(F'(x)=\lim_{h\to 0}\frac{F(x+h)-F(x)}{h}\Longrightarrow \lim_{h\to 0}\frac{f(x+h)+g(x+h)-(f(x)+g(x))}{h}=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}+\lim_{h\to 0}\frac{g(x+h)-g(x)}{h}\)
Which implies
\(F'(x)=f'(x)+g'(x)\)

Products

\(F(x)=f(x)\cdot g(x)\Longrightarrow F'(x)=f'(x)g(x)+f(x)g'(x)\)

Proof

\(F'(x)=\lim_{h\to 0}\frac{F(x+h)-F(x)}{h}\Longrightarrow \lim_{h\to 0}\frac{f(x+h)\cdot g(x+h)-f(x)g(x)}{h}\)
We now add and subtract \(f(x)g(x+h)\) to the numerator
\(F'(x)=\lim_{h\to 0}\frac{f(x+h)\cdot g(x+h)-f(x)g(x+h)+f(x)g(x+h)-f(x)g(x)}{h}\Longrightarrow\lim_{h\to 0}\frac{[f(x+h)-f(x)]g(x+h)+f(x)[g(x+h)-g(x)]}{h}\)
Rearranging yields
\(F'(x)=\lim_{h\to 0}g(x+h)\frac{f(x+h)-f(x)}{h}+f(x)\frac{g(x+h)-g(x)}{h}\)
This implies
\(F'(x)=g(x)f'(x)+f(x)g'(x)\)

Example

Find the derivative of 
\(h(x)=(x^3-x)(5x^4+x^2)\)

Quotients

\(F(x)=\frac{f(x)}{g(x)}\Longrightarrow F'(x)=\frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}\)

Examples

  1. \(F(x)=\frac{3x-5}{x-2}\)
  2. Find the derivative for the average cost function \(AC=\frac{C(x)}{x}\)