Edexcel IAL - Pure Maths 3- 4.3 The use of dy/dx = 1 / (dx/dy)- Study notes - New syllabus
Edexcel IAL – Pure Maths 3- 4.3 The use of dy/dx = 1 / (dx/dy) -Study notes- New syllabus
Edexcel IAL – Pure Maths 3- 4.3 The use of dy/dx = 1 / (dx/dy) -Study notes -Edexcel A level Physics – per latest Syllabus.
Key Concepts:
- 4.3 The use of dy/dx = 1 / (dx/dy)
Using the Reciprocal Derivative
Sometimes a function is given with \( x \) in terms of \( y \) instead of \( y \) in terms of \( x \).
In such cases, it may be easier to:
- Differentiate \( x \) with respect to \( y \)
- Then use the reciprocal relationship to find \( \dfrac{dy}{dx} \)
Reciprocal Derivative Formula
If:
\( \dfrac{dx}{dy} \neq 0 \)
then:
\( \dfrac{dy}{dx} = \dfrac{1}{\left(\dfrac{dx}{dy}\right)} \)
This follows from the chain rule.
When This Method Is Used
- When \( x \) is given explicitly as a function of \( y \)
- When rearranging for \( y \) is difficult or unnecessary
- Common in trigonometric and implicit-type questions
Method
- Differentiate \( x \) with respect to \( y \)
- Find \( \dfrac{dx}{dy} \)
- Take the reciprocal to obtain \( \dfrac{dy}{dx} \)
- Simplify the final expression
Example
Given:
\( x = y^3 \)
Find \( \dfrac{dy}{dx} \).
▶️ Answer / Explanation
Differentiate with respect to \( y \):
\( \dfrac{dx}{dy} = 3y^2 \)
Take the reciprocal:
\( \dfrac{dy}{dx} = \dfrac{1}{3y^2} \)
Example
Given:
\( x = e^{2y} \)
Find \( \dfrac{dy}{dx} \).
▶️ Answer / Explanation
Differentiate with respect to \( y \):
\( \dfrac{dx}{dy} = 2e^{2y} \)
Take the reciprocal:
\( \dfrac{dy}{dx} = \dfrac{1}{2e^{2y}} \)
Example
Given:
\( x = \sin 3y \)
Find \( \dfrac{dy}{dx} \).
▶️ Answer / Explanation
Differentiate with respect to \( y \):
\( \dfrac{dx}{dy} = 3\cos 3y \)
Use the reciprocal formula:
\( \dfrac{dy}{dx} = \dfrac{1}{3\cos 3y} \)
\( = \dfrac{1}{3}\sec 3y \)