Section 3

Rules of Differentiation

Differentiation of a function \(f(x)\) with respect to \(x\) is the process to obtain the derivative of the function. Below we list some frequently used rules of differentiation in economics:

• Differentiation of a constant term is zero.

  Example 1:

  Suppose \(y = f(x) = 3x + 2\)

  $$\frac{dy}{dx} = {f}'(x) = \frac{d}{dx}(3x) + \frac{d}{dx}(2) = \frac{d}{dx}(3x) + 0$$

  
  
  
  
• The derivative of the linear function \(y = a + \;\)\(b\)\(x\) is equal to \(b\), the coefficient of \(x\). The derivative of the function \(y = a + bx\), is \(\frac{dy}{dx} = b\), is also the slope of the function.

  Example 2:

  Suppose \(y = f(x) = 3x + 2\) $$\frac{dy}{dx} = {f}'(x) = \frac{d}{dx}(3x) + \frac{d}{dx}(2) = 3 + 0 = 3$$

  
  
  
  
• The Power Function Rule (We need to apply this rule frequently in economics.)

  If \(y = f(x) = kx^{\color{red}{n}}\), where \(k\) is the coefficient, \(\color{red}{n}\) is the exponent and \(n \neq 0\), then the derivative is:

  \(\frac{dy}{dx} = (coefficient) \cdot (exponent) \cdot x^{exponent - 1} = k \cdot \color{red}{n} \cdot x^{\color{red}{n - 1}}\)

  Example 3:

  Given \(f(x) = 4x^{2}\),           \({f}'(x) = 4 \cdot 2 \cdot x^{2 - 1} = 8x\)
  Given \(f(x) = 4x^{2} + 10\),       \({f}'(x) = 4 \cdot 2 \cdot x^{2 - 1} + \frac{d}{dx}(10) = 8x + 0 = 8x\)
  Given \(f(x) = 4x^{2} + 5x + 10\),     \({f}'(x) = \frac{d}{dx}4x^{2} + \frac{d}{dx}5x + \frac{d}{dx}(10) = 8x + 5 + 0\)

  
  
  
  
• The Product Rule:

  The derivative of a product of two functions, \(f(x) = \color{red}{g(x)} \cdot \color{#42a1f4}{h(x)}\) where both \(g(x)\) and \(h(x)\) are differentiable functions, is:

  \(f'(x) = \color{red}{g(x)} \cdot \color{#42a1f4}{{h}'(x)} + \color{#42a1f4}{h(x)} \cdot \color{red}{{g}'(x)}\)

  Example 4

  Given \(f(x) = \color{red}{(x^{2} + 2)} \color{#42a1f4}{(2x - 5)}\)
  Where, \(\color{red}{g(x) = (x^{2} + 2)}\) and \(\color{#42a1f4}{h(x) = (2x - 5)}\)

  \({f}'(x) = \color{red}{(x^{2} + 2)} \frac{d}{dx} \color{#42a1f4}{(2x - 5)} + \color{#42a1f4}{(2x - 5)} \frac{d}{dx} \color{red}{(x^{2} + 2)}\) $$= (x^{2} + 2)(2) + (2x - 5)(2x) \qquad\quad\;\;$$ $$= 2x^{2} + 4 + 4x^{2} - 10x \qquad\qquad\qquad$$ $$= 6x^{2} - 10x + 4 \qquad\qquad\qquad\qquad\;\;$$

  
  
  
  
• The Quotient Rule

  The derivative of a quotient \(f(x) = \frac{\color{red}{g(x)}}{\color{#42a1f4}{h(x)}}\), where both \(g(x)\) and \(h(x)\) are differentiable functions and \(h(x)\) never equals zero, is: $$\frac{[(the \; denominator) \cdot (differentiation \; of \; the \; numerator) - (the \; numerator) \cdot (differentiation \; of \; the \; denominator)]}{square \; of \; the \; denominator}$$ $${f}'(x) = \frac{\color{#42a1f4}{h(x)} \color{red}{{g}'(x)} - \color{red}{g(x)}\color{#42a1f4}{{h}'(x)}}{\color{#42a1f4}{(h(x))^{2}}}$$

  Example 5:

  \(f(x) = \frac{\color{red}{x + 1}}{\color{#42a1f4}{x - 1}}\) $${f}'(x) = \frac{\color{#42a1f4}{(x - 1)} \frac{d}{dx} \color{red}{(x + 1)} - \color{red}{(x + 1)} \frac{d}{dx} \color{#42a1f4}{(x - 1)}}{\color{#42a1f4}{}(x - 1)^{2}}$$ $$= \frac{(x - 1)(1) - (x + 1)(1)}{(x - 1)^{2}} \qquad$$ $$= \frac{x - 1 - x - 1}{(x - 1)^{2}} = -\frac{2}{(x - 1)^{2}}$$

  
  
  
  
• The Generalized Power Function Rule

  The derivative of a function raised to a power, \(f(x) = [g(x)]^{n}\), where \(g(x)\) is a differentiable function, is: $${f}'(x) = \color{red}{n}[g(x)]^{\color{red}{n - 1}} \cdot {g}'(x)$$

  Example 6: Given \(f(x) = [x^{2} + 2]^{\color{red}{3}}\) $${f}'(x) = \color{red}{3}[x^{2} + 2]^{\color{red}{3 - 1}} \frac{d}{dx}(x^{2} + 2)$$ $$= 3[x^{2} + 2]^{2} \cdot (2x) \;\;\;$$ $$= 6x[x^{2} + 2]^{2} \qquad \;\;\;$$

  
  
  
  
• The Natural Exponential Function Rule and the Natural Logarithmic Function Rule

  Given \(f(x) = e^{g(x)}\), where \(g(x)\) is a differentiable function, the derivative is: $${f}'(x) = e^{g(x)} \cdot {g}'(x)$$

  Given \(f(x) = \ln{(g(x))}\), where \(g(x)\) is a strictly positive differentiable function, the derivative is: $${f}'(x) = \frac{1}{g(x)} \cdot {g}'(x)$$

  Example 7: Given \(f(x) = e^{(x^{2} + 2)}\) $${f}'(x) = e^{(x^{2} + 2)} \cdot \frac{d}{dx}(x^{2} + 2)$$ $$= e^{(x^{2} + 2)}(2x) \quad\;$$ $$= 2xe^{(x^{2} + 2)} \quad\;\;\;$$

  
  

  Example 8: Given \(\ln{(x^{2} + 2)}\) $${f}'(x) = \frac{1}{x^{2} + 2}\frac{d}{dx}(x^{2} + 2)$$ $$= \frac{2x}{x^{2} + 2} \qquad\;\;$$