Differentiation (Single Variable) and Economic Applications

1) Derivative: meaning

The derivative measures the rate of change of a function.

If $y=f(x)$, the derivative is:

Economic meaning: “marginal” change.

• Marginal cost = rate of change of cost w.r.t. output
• Marginal utility = rate of change of utility w.r.t. consumption

2) Rules of differentiation (must-know)

1. Power rule:

2. Constant:

3. Constant multiple:

4. Sum/difference:

5. Product rule:

6. Quotient rule:

7. Chain rule:

Key standard derivatives:

• $\frac{d}{dx}(e^x)=e^x$
• $\frac{d}{dx}(\ln x)=\frac{1}{x}$
• $\frac{d}{dx}(a^x)=a^x\ln a$
• $\frac{d}{dx}(\ln f(x))=\frac{f'(x)}{f(x)}$