# Implicit Differentiation and Related Rates (HL) | Maths with Lemon

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### Implicit Differentiation & Related Rates

IB Mathematics: Analysis and Approaches Higher Level · AHL 5.14 · Lesson 10.1. Differentiate equations involving both variables, connect changing quantities through time, and solve geometric rate problems with correct units and signs.

Lesson 10.1 worksheet (PDF) AA Calculus overview Differentiation rules


#### 1. The Idea Behind Implicit Differentiation


##### What you need to know

- Explicit functions are written as \(y=f(x)\).
- An implicit relation may mix \(x\) and \(y\), such as \(x^2+y^2=4\).
- When differentiating terms involving \(y\), use the chain rule.

##### Key points

Treat \(y\) as a function of \(x\). Differentiate both sides with respect to \(x\).

Differentiate the equation \(x^2+y^2=4\).

The gradient is undefined where \(y=0\), corresponding to vertical tangents on the circle.


#### 2. Differentiating Implicit Equations


##### From your worksheet · Page 2

- Differentiate each side term by term.
- Collect terms containing \(\frac{dy}{dx}\).
- Factor and isolate the derivative.
- Keep your answer in terms of \(x\) and \(y\), unless asked otherwise.

##### Step-by-step example

Step 1: Differentiate with respect to \(x\).

Step 2: Make the derivative the subject.


#### 3. Finding the Gradient at a Point


##### From your worksheet · Page 2

- Differentiate the entire relation first.
- Substitute the given coordinates into the derivative.
- Check that the point satisfies the original relation.
- Use the gradient to write a tangent or normal equation if required.

##### Worked example

Differentiate implicitly:

Substitute \(x=-1\), \(y=2\):

Since \(3(-1)^3+2(2)^2=5\), the point lies on the curve.


#### 4. Related Rates: The Four-Step Method


##### What you need to know

- Rates describe change with respect to time \(t\).
- Increasing quantities have positive rates; decreasing quantities have negative rates.
- Build an equation connecting the quantities before differentiating.
- Keep units consistent: length/time, area/time or volume/time.

##### Method

- Sketch and label the quantities that change.
- Write a geometric or algebraic relationship.
- Differentiate both sides with respect to \(t\).
- Substitute the instantaneous values and solve, including units.

#### 5. Rectangles and Right Triangles


##### Worksheet · Pages 3–4

- Rectangle area and perimeter.
- Right-triangle area.
- Changing hypotenuse via Pythagoras.
- Negative rates represent shrinking lengths.

##### Essential formulas

At an instant, \(l=7\) cm and \(w=1\) cm. The length increases at 3 cm/s and the width decreases at 5 cm/s.

The area is decreasing at that instant.


#### 6. Circles, Spheres, Cylinders and Cones


##### Worksheet · Pages 5–6

- Differentiate area and volume formulas with respect to time.
- For cylinders and cones, both radius and height may change.
- If volume is constant, set \(\frac{dV}{dt}=0\).
- Express answers in the correct squared or cubed units per time.

##### Formula collection

A sphere has radius 8 cm and the radius decreases at 7 cm/min.


#### 7. Interactive Related-Rates Calculator


##### Explore how rates interact

- Choose a rectangle or circle.
- Enter the dimensions at a particular instant.
- Enter positive or negative rates of change.
- Compare the calculated rate with your own differentiation.

##### Instantaneous rate explorer

Use a consistent length unit and time unit (for example cm and seconds).


#### 8. Multi-Equation Related Rates and HL Practice


##### Worksheet · Pages 7–8

- Connect area, perimeter and side length of squares.
- Connect volume, surface area and side length of cubes.
- Relate sphere volume to its surface area.
- Use the chain rule across multiple equations.

##### Worked example: changing cube

Start with \(V=s^3\) and \(S=6s^2\).

Open the full Lesson 10.1 worksheet


#### 9. Revision Checklist


##### Before your assessment

- Apply the chain rule to expressions involving \(y\).
- Find derivatives and gradients at specified points.
- Differentiate with respect to time in related-rates problems.
- Use product rule when two changing dimensions are multiplied.
- Interpret signs, units and physical constraints.

##### Common mistakes to avoid

Do not forget \(\frac{dy}{dx}\) when differentiating terms in \(y\).

Do not replace changing variables by their instantaneous values before differentiation.


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