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MasterMaths online tuition: A-Level Maths Implicit Differentiation With Worked Examples

A-Level Maths Implicit Differentiation With Worked Examples

Writer: Ryan Harvey
Ryan Harvey
Oct 12
4 min read

Implicit differentiation is needed when x and y are linked by an equation that is not written as y = f(x). The key idea is to treat y as a function of x, so differentiating a term such as y² requires the chain rule. This A-Level Maths guide works through the method and explains where students commonly lose marks.



When do you use implicit differentiation?


Consider the circle x² + y² = 25. Solving for y would require y = ±√(25 − x²), giving upper and lower branches. Implicit differentiation handles the original equation directly and can be much more convenient for tangents and related-rates questions. The method also applies to products such as xy or powers involving x and y together. Each time you differentiate a y expression with respect to x, remember that y changes as x changes.



Differentiate a circle step by step


Start from x² + y² = 25. Differentiate both sides with respect to x: 2x + 2y(dy/dx) = 0. The derivative of y² is 2y(dy/dx), not just 2y, because of the chain rule. Rearrange to get dy/dx = −x/y, provided y is not zero. At the point (3,4), the gradient is −3/4. The point is on the circle because 3² + 4² = 25. At y = 0, the vertical tangent requires separate interpretation.



Understand why the chain rule appears


Suppose y is a function of x and we differentiate y³. Think of y as an intermediate quantity: the derivative of u³ is 3u² and du/dx becomes dy/dx. Thus d(y³)/dx = 3y²(dy/dx). The extra derivative is not an optional decoration; it accounts for the change in y with x. Repeating this explanation is more reliable than memorising isolated forms without understanding the underlying function composition.



Differentiate an equation containing xy


For xy + y² = 10, differentiating xy needs the product rule: d(xy)/dx = x(dy/dx) + y. Meanwhile d(y²)/dx = 2y(dy/dx). Therefore x(dy/dx) + y + 2y(dy/dx) = 0. Collect derivative terms to obtain (x + 2y)(dy/dx) = −y, and hence dy/dx = −y/(x + 2y), wherever the denominator is nonzero. Forgetting the product-rule term y is a frequent mistake.



Use the derivative to form a tangent equation


Return to x² + y² = 25 at (3,4). The gradient is −3/4, so a tangent has equation y − 4 = (−3/4)(x − 3). This point-gradient form is perfectly valid, and multiplying out is optional unless requested. To check it conceptually, the circle’s radius from the origin to (3,4) has gradient 4/3, and the tangent is perpendicular with gradient −3/4. Linking calculus to geometry makes the result more memorable.



Common implicit differentiation exam mistakes


Students sometimes differentiate y² as 2y and omit dy/dx, treat xy as a simple product of derivatives, or substitute coordinates into the gradient before rearranging correctly. Another trap is cancelling expressions that might equal zero without stating a restriction. Write the derivative terms carefully, collect dy/dx once, then substitute any specific point. If the required answer is a tangent or normal, check whether the gradient needs a negative reciprocal and whether the supplied point satisfies the original curve.



Frequently asked questions about A-Level Maths implicit differentiation


Is implicit differentiation part of A-Level Pure Maths?

Yes, it forms part of advanced differentiation work; check the exact coverage of your exam board.


What should I revise first?

The chain rule, product rule, rearranging algebra and equations of straight-line tangents are essential foundations.


Explore connected maths learning topics


The following carefully selected lessons and articles support the wider skills mentioned above. Use them to find the topic that is most relevant to your current difficulty, rather than trying to read every resource at once.



Book a free online maths introduction


If your child needs a clearer explanation or you want to discuss whether one-to-one online maths tuition would be useful, MasterMaths offers an introductory conversation. Share the year group, any recent feedback and one or two questions that have caused difficulty. Ryan can explain how interactive online tuition works and help you decide on an appropriate next step. There is no promise of a fixed grade outcome; the aim is confident, independent understanding.


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