DIFFERENTIATION OF ALGEBRAIC FUNCTIONS

CONTENT

1. Rules of differentiation such as: (i) sum and difference (ii) chain rule (iii) product rule (iv)quotient rule.
2. Application to real life situation such as maxima and minima, velocity, acceleration and rate of change etc

RULES OF DIFFERENTIATION

SUM AND DIFFERENCE RULE

(a) If \(y = u + v \\ \frac{dy}{dx} = \frac{d(u + v)}{dx} \\ \frac{dy}{dx} = \frac{du}{dx} + \frac{dv}{dx}\)

(sum rule)

(b) If \(y = u -v \\ \frac{dy}{dx} = \frac{d(u -v)}{dx} \\ \frac{dy}{dx} = \frac{du}{dx} -\frac{dv}{dx}\\\)

(difference rule)

Example 1: Differentiate the following with respect to \(x\)

(a) \(y = x^4 + 4x^3 -2x + 1 \)

(b) \( y = 6x^3 –x^2 -4x + 1\)

SOLUTION

\(y = x^4 + 4x^3 -2x + 1 \\ \frac{dy}{dx} = 4x^3 + 12x^2 -2 \\ ∴ \frac{dy}{dx} = 4x^3 + 12x^2 -2\)

(a) \(y = x^4 + 4x^3 -2x + 1 \\ \frac{dy}{dx} = 4x^3 + 12x^2 -2 \\ ∴ \frac{dy}{dx} = 4x^3 + 12x^2 -2\)

(b) \(y = 6x^3 –x^2 -4x + 1 \\ \frac{dy}{dx} = 18x^2 -2x -4 \\ ∴ \frac{dy}{dx} = 18x^2 -2x -4\)

EXAMPLE 2:

Find \(\frac{dy}{dx}\) of the equation of the curve \(x^3 + 3x^2 -9x + 5\)

SOLUTION

\(y = x^3 + 3x^2 -9x + 5 \\ \frac{dy}{dx} = 3x^2 + 6x -9 \\ ∴ \frac{dy}{dx} = 3x^2 + 6x -9\)

CHAIN RULE (Function of a Function)

The chain rule is used to find the derivatives of functions that have powers.