Differentiation and its Applications

A technique used when y cannot

A technique used when y cannot be explicitly expressed as a function of x. We differentiate both sides of the equation with respect to x, treating y a

When both x and y

When both x and y are expressed in terms of a parameter t, we use dy/dx = (dy/dt)/(dx/dt). For example, if x = t², y = 2t, then dy/dx = (2)/(2t) = 1/t

Used for functions of the form

Used for functions of the form [f(x)]^g(x). Take natural log of both sides, then differentiate. For y = x^x, log y = x log x, so (1/y)(dy/dx) = log x

Second derivative d²y/dx² is the derivative

Second derivative d²y/dx² is the derivative of dy/dx, third derivative d³y/dx³ is the derivative of d²y/dx², and so on. These represent rates of chang

Cost function C(x) = Variable cost

Cost function C(x) = Variable cost + Fixed cost. Revenue function R(x) = price × quantity. Marginal cost MC = dC/dx and marginal revenue MR = dR/dx re