MAT 1300 Lecture Notes - Lecture 6: Implicit Function, Power Rule

• School: University of Ottawa
• Department: Mathematics
• Course: MAT 1300
• Professor: All

Document Summary

The chain rule: let f (x) and g(x) be di erentiable functions. The general power rule: let g(x) be a di erentiable function and let h(x) = [g(x)]n. then, h(cid:48)(x) = n[g(x)]n 1g(cid:48)(x). Ex: given r(z) = (z2 3z)5, nd r(cid:48)(z). Ex: given that h(x) = x(5x + 1)3, nd h(cid:48)(x). Ex: find g(cid:48)(2) given that g(x) = (x2 + 4)1/3 x. Ex: find the equation of the tangent line to f (x) = 5 (2x 1)7 at the point x = 1. Ex: the cost of producing x units of a product is given by c(x) = where c(x) is the cost in dollars. Find the marginal cost for producing 20 units. Many relationships can be expressed explicitly in the form y = f (x), but some can only be written implicitly as a mix of x"s and y"s on either side of the equality sign. Ex: x2 5y4 + 3y + 2x = 10.