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CIE A-Level Maths Study Notes

2.4.3 Parametric and Implicit Differentiation

Differentiation enables students to analyze how functions behave dynamically. This section specifically explores the sophisticated techniques of parametric and implicit differentiation, which are vital for examining more complex functions and equations.

Parametric Equations

Parametric equations are a set of functions where xx and yy are both described in terms of a third variable, often denoted as tt. This method is particularly useful for characterizing curves that do not lend themselves to simple xyx-y functional relationships.

parametric equations

Image courtesy of Mathbooks

For parametric equations where xx and yy are functions of tt:

1. Differentiate x(t)x(t) and y(t)y(t) with respect to tt.

2. Calculate dydx\frac{dy}{dx} as dydtdxdt\frac{\frac{dy}{dt}}{\frac{dx}{dt}}

Example:

Differentiate x=te2tx = t - e^{2t} and y=t+e2ty = t + e^{2t}:

Solution:

dxdt=12e2t,dydt=1+2e2t\frac{dx}{dt} = 1 - 2e^{2t}, \quad \frac{dy}{dt} = 1 + 2e^{2t}

Thus,

dydx=1+2e2t12e2t\frac{dy}{dx} = \frac{1 + 2e^{2t}}{1 - 2e^{2t}}

Simplified:

dydx=2e2t+12e2t1\frac{dy}{dx} = -\frac{2e^{2t} + 1}{2e^{2t} - 1}

Implicit Functions

Implicit functions are those in which xx and yy are intermingled within the same equation. They often describe complex shapes and relationships that are not easily separated into distinct functions.

Implicit differentiation for equations with xx and yy mixed:

1. Treat y as a function of xx.

2. Differentiate the whole equation with respect to xx.

3. Solve for dydx\frac{dy}{dx}.

Example:

Differentiate the implicit function $x^2 + y^2 = xy + 7 withrespectto with respect to xtofind to find \frac{dy}{dx}.</p><p></p><p><strong>Solution:</strong></p><p>Differentiateeachtermwithrespectto.</p><p></p><p><strong>Solution: </strong></p><p>Differentiate each term with respect to x,applyingtheproductruletotheterm, applying the product rule to the term xy:</p><p></p>:</p><p></p>2x + 2yy' = y + xy'<p></p><p>Here,<p></p><p>Here, y'denotes denotes \frac{dy}{dx}.Rearrangeandsolvefor. Rearrange and solve for y':</p><p></p>:</p><p></p>y' = y - 2x$

Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
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Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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