# MCV4U – Grade 12 Calculus & Vectors – Derivatives

Grade 12 – Calculus and Vectors

Derivatives Test

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 be 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