9th Lecture

Chain Rule

Whenever you have h(u(x)), to take the derivative, you take the derivative first of h'(u(x)), then you multiply that by u'(x). So the derivative of a quantity of the type h, you know, and then you have some quantity is derivative of the of f evaluated at that quantity times the derivative of the quantity.

Example

f(x) = 3x + 1^{0.1}

h(x) = x^{0.1}

u(x) = 3x + 1

f'(x) = h'(u(x)) * u'(x)

h'(x) = 0.1 * x^{-0.9}

u'(x) = 3

f'(x) = 0.1 * (3x + 1)^{-0.9} * 3 = 0.3 * (3x + 1)^{-0.9}

Generalized Power Rule

Whenever you want to take the derivative of u(x)^n, that would always be n * u(x)^{n-1} * u'(x).

Example

Find the derivative of \sqrt{3x + 1}.

Rewrite as (3x + 1)^{1/2}, then apply the generalized power rule.

(1/2) * (3x + 1)^{-1/2} * 3 = (3/2) * (3x + 1)^{-1/2}

Chain Rule for Marginal Product

Chain rule can be used to find the marginal product \frac{dP}{dN}, where P is a function of Q, and Q is a function of N.

\frac{dP}{dN} = \frac{dP}{dQ} * \frac{dQ}{dN}