Differentials

Definition

Differentials provide an approximation of how a function changes as its input changes. They are used to approximate small changes in functions using the derivative.

5 Must Know Facts For Your Next Test

1. Differentials can be expressed as $dy = f'(x)dx$, where $dy$ is the differential change in $y$ and $dx$ is the differential change in $x$.
2. The differential $dx$ is often treated as an independent variable representing a small change in $x$.
3. Differentials are crucial for linear approximations, which estimate the value of a function near a given point using its tangent line.
4. In practical applications, differentials help estimate errors and changes in measurements.
5. The notation for differentials aligns with Leibniz's notation for derivatives, making them easier to understand within the context of calculus.

Review Questions

• What is the formula for expressing differentials?
• How do differentials assist in linear approximations?
• Why are differentials important for estimating errors?

Related terms

Derivative: A measure of how a function changes as its input changes; represented as $f'(x)$ or $\frac{dy}{dx}$.

Linear Approximation: An estimation method using the tangent line to approximate the value of a function near a given point.

Tangent Line: A straight line that touches a curve at one point and has the same slope as the curve at that point.