The Derivative

Average rate of change, then let the change shrink: \[\frac{d y}{d x} = f'(x)= \lim_{\Delta x \rightarrow 0} \frac{f\left(x+\Delta x\right)-f\left(x\right)}{\Delta x}\]

Use the limit definition to find \(f'(x)\) for \(y = f(x) = x^2 + 3x\).

Limits and Continuity

\(\lim_{x \rightarrow a} f(x)\) exists only if the left-side and right-side limits agree; \(f\) is continuous at \(a\) if the limit exists and equals \(f(a)\).

Continuity and Differentiability

Continuity is necessary for differentiability, but not sufficient.

\[g(x) = \begin{cases} 2x & \text{if } x < 1 \\ x + 1 & \text{if } x \geq 1 \end{cases}\]

Plot \(g\). Is it continuous at \(x = 1\)? Is it differentiable there?

The Rules

Product, Quotient, and Chain Rule:

\[\frac{d}{d x}[f g]=f g' + f' g \qquad \frac{d}{d x} \frac{f}{g}=\frac{f' g-f g'}{g^2} \qquad \frac{d z}{d x}=\frac{d z}{d y} \cdot \frac{d y}{d x}\]

Find \(dy/dx\):

• \(y = (x^2 + 1)(3x - 2)\)

• \(y = \dfrac{x}{x + 2}\)

• \(y = (2x + 1)^3\)

Exponential and Log Functions

\[\frac{d}{dx} e^{x} = e^{x} \qquad\qquad \frac{d}{dx} \ln x = \frac{1}{x}\]

For \(e^{f(x)}\) and \(\ln f(x)\), combine these with the chain rule. Find \(dy/dx\):

• \(y = 3e^{2x}\)

• \(y = x^2 \ln x\)

• \(y = \ln(x^2 + 4)\)

Elasticity and Logs

\[\varepsilon = \frac{dy}{dx} \cdot \frac{x}{y} = \frac{d \ln y}{d \ln x}\]

Demand is \(Q = \dfrac{50}{p^{2}}\). What is the price elasticity of demand? Is demand elastic or inelastic?

Partial Derivatives

Differentiate with respect to one variable, holding the others constant.

For \(f(x, y) = x^2 y + y^3\), find \(f_x\), \(f_y\), and the gradient \(\nabla f\). Is \(f_x\) increasing in \(y\)?

Production Function

\[Q = K^{0.3} L^{0.7}\]

Find the marginal product of capital \(Q_K\) and of labor \(Q_L\).

Is the marginal product of labor decreasing in \(L\)?

Total Differential and Total Derivative

For \(y = f(x_1, x_2, \ldots, x_n)\), the total differential is \[dy = f_1 \cdot dx_1 + f_2 \cdot dx_2 + \cdots + f_n \cdot dx_n\]

If the other variables depend on \(x_1\), the total derivative with respect to \(x_1\) is \[\frac{dy}{dx_1} = f_1 + f_2 \cdot \frac{dx_2}{dx_1} + \cdots + f_n \cdot \frac{dx_n}{dx_1}\]

Example

\[z = x^2 + xy, \quad \text{where } x = t \text{ and } y = t^2\]

Find the total differential \(dz\) and the total derivative \(dz/dt\).

Implicit-Function Rule

For \(F(x, y) = 0\), totally differentiate: \(F_x \cdot dx + F_y \cdot dy = 0\), so \[\frac{dy}{dx} = -\frac{F_x}{F_y}\]

Market Equilibrium

Demand: \(Q^d = Y - P^2\), with \(Y\) income

Supply: \(Q^s = P + \ln P\)

The equilibrium condition cannot be solved for \(P\) explicitly. Write it as \(F(P, Y) = 0\) and find \(\partial P / \partial Y\). Does higher income raise the price?