Further Differentiation and Applications

CIE A-Level Further Mathematics· Unit 2: Further Differentiation and Applications· 25 min read

1. Advanced Differentiation Techniques★★★☆☆⏱ 10 min

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Beyond basic differentiation of explicit functions, we extend techniques to relations defined implicitly and curves given by parametric equations, which are common in CIE exam questions.

📘 Definition

Parametric Differentiation

$x = x (t), y = y (t)$

The derivative of $y$ with respect to $x$ is found by dividing the derivative of $y$ with respect to $t$ by the derivative of $x$ with respect to $t$ :

Example:

For $x = t^{2}, y = sin t$ , $\frac{d y}{d x} = \frac{c o s t}{2 t}$

📐 Worked Example

Find $\frac{d y}{d x}$ for the ellipse defined parametrically by $x = 2 cos θ$ , $y = 3 sin θ$ , at the point where $θ = \frac{π}{4}$ .

1
First differentiate $x$ and $y$ individually with respect to $θ$ :
2
$\frac{dx}{d\theta} = -2\sin\theta, \quad \frac{dy}{d\theta} = 3\cos\theta$
3
Apply the parametric differentiation formula $\frac{d y}{d x} = \frac{d y / d θ}{d x / d θ}$ :
4
$\frac{dy}{dx} = \frac{3\cos\theta}{-2\sin\theta} = -\frac{3}{2}\cot\theta$
5
Substitute $θ = \frac{π}{4}$ , where $cot (\frac{π}{4}) = 1$ :
6
$\frac{dy}{dx}\bigg|_{\theta=\frac{\pi}{4}} = -\frac{3}{2}}]}}},{$

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