University:
Virtual High School
Course:
MCV4U | Calculus and Vectors
Academic year:

2023
Views:

252

Pages:

2
Author:

anon

MCV4U – Grade 12 Calculus & Vectors – Derivatives Derivatives of a Polynomial Function ● Derivative rules simplify the process of differentiating polynomials with first principles ● When differentiating radicals, we rewrite radicals in fraction exponent form ○ ie root x = x^1/3 ● To differentiate a power of x that is in the denominator, first express it as a power with a negative exponent ● Derivative rule: xn = nxn-1 ● Sum of derivatives: (f(x) + g(x))’= f’(x) + g’(x) ● Difference of derivatives: (f(x) – g(x))’ = f’(x) – g’(x) Product Rule ● Product Rule: (f(x) * g(x)) = f’(x)g(x) + f(x)g’(x) ● Leibniz Notation: d/dx[f(x)g(x)] = d/dx[f(x)]g(x) + f(x) d/dx[g(x)] Velocity, Acceleration, and Second Derivatives ● The second derivative of a function is determined by differentiating the first derivative of the function ● For a given position function s(t), its velocity function is v(t), or s’(t), and its acceleration is a(t), v’(t), or s’’(t) ● When v(t) = 0, the object is at rest. There are many instances when an object will temporarily be at rest when changing directions. ● When v(t) > 0, the object is moving in the positive direction ● When v(t) < 0, the object is moving in the negative direction ● When a(t) > 0, the velocity of the object is increasing ● When a(t) < 0, the velocity of the object is decreasing ● An object is speeding up if a(t) x v(t) > 0 and slowing down if v(t) x a(t) < 0. Chain Rule ● Used to differentiate composite functions, f = g o h. ● Given a function, the Chain rule is: ○ (f(g(x)))’ = f’(g(x) * g’(x) ● In Leibniz notation, ○ dy/dx = dy/du * du/dx Quotient Rule ● To find the derivative of a quotient: ○ q(x) = f’(x)g(x) – f(x)g’(x) / g2(x) Rate of Change problems ● Demand or price function p(x) is the price at which x units of a product or service can be sold ● Revenue function R(x) is the total revenue from the sale of x units of a product or service. R(x) = x * P(x) ● Cost function, C(x), is the total cost of producing x units of a product or service ● Profit function, P(x) is P(x) = R(x) – C(x) ● C’(x) is the marginal cost function ● R’(x) is the marginal revenue function ● P’(x) is the marginal profit function