Differentials
Use dy = f′(x)dx for small changes.
Key idea
Use dy = f′(x)dx for small changes.
In depth
dy = f′(x) dx estimates the change in y produced by a small change dx. It formalises linear approximation and is useful for estimating errors in measurements.
Worked example
If a side of a square is 10 ± 0.1, then dA = 2(10)(0.1) = 2, so area is 100 ± 2.
Real-world uses
- Estimating the error in a measured volume from a small error in its radius
Common mistake
Confusing dy (tangent estimate) with Δy (true change).
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Differentials?
Use dy = f′(x)dx for small changes. It is a topic at the expert level in Calculus.
Where is Differentials used in real life?
Real-world examples: Estimating the error in a measured volume from a small error in its radius.
What should I learn before Differentials?
Make sure you are comfortable with Linear Approximation, the previous topic in Calculus.
What is a common mistake with Differentials?
Confusing dy (tangent estimate) with Δy (true change).
How can I practise Differentials?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Differentials?
Move on to Critical Points next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Second Derivative Test – f″ > 0 means concave up; f″ < 0 concave down.