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.

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