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A Level Mathematics Syllabus Statement

Differentiation

Syllabus Content

Differentiate using the product rule, the quotient rule and the chain rule, including problems involving connected rates of change and inverse functions

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Furthermore

Examples

1. Chain Rule:
The chain rule is used when differentiating composite functions. If we have a function \( y = f(u) \) and \( u = g(x) \), then the derivative of \( y \) with respect to \( x \) is given by:

$$ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} $$

Example: Differentiate \( y = \sin(3x^2) \) with respect to \( x \).

Let \( u = 3x^2 \). Then, \( \frac{du}{dx} = 6x \) and \( \frac{dy}{du} = \cos(u) \). Using the chain rule:

$$ \frac{dy}{dx} = \cos(3x^2) \cdot 6x = 6x \cos(3x^2) $$

2. Product Rule:
The product rule is used when differentiating the product of two functions. If \( y = u \cdot v \) where both \( u \) and \( v \) are functions of \( x \), then the derivative of \( y \) with respect to \( x \) is:

$$ \frac{dy}{dx} = u' \cdot v + u \cdot v' $$

Example: Differentiate \( y = x^2 \cdot \ln(x) \) with respect to \( x \).

Using the product rule:

$$ \frac{dy}{dx} = 2x \cdot \ln(x) + x^2 \cdot \frac{1}{x} = 2x \ln(x) + x $$

3. Quotient Rule:
The quotient rule is used when differentiating the quotient of two functions. If \( y = \frac{u}{v} \) where both \( u \) and \( v \) are functions of \( x \) and \( v \neq 0 \), then the derivative of \( y \) with respect to \( x \) is:

$$ \frac{dy}{dx} = \frac{u' \cdot v - u \cdot v'}{v^2} $$

Example: Differentiate \( y = \frac{x^2}{\sin(x)} \) with respect to \( x \).

Using the quotient rule:

$$ \frac{dy}{dx} = \frac{2x \cdot \sin(x) - x^2 \cdot \cos(x)}{\sin^2(x)} $$

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