AP Calculus AB and BC glossary

Differential

Also called: dx, dy

The differential dy is the change in output along the tangent line corresponding to a small change dx in the input. It approximates the true change in the function and is the notation that makes substitution and separable equations work.

dy = f'(x)\,dx

The distinction worth holding onto is between $\Delta y$ , the exact change in the function, and $dy$ , the change along the tangent line. They agree closely when $dx$ is small, which is exactly the claim linearization makes.

Differentials are also why u-substitution is written the way it is. Setting $u = g(x)$ gives $du = g'(x)\,dx$ , which is what lets you replace one differential with another inside an integral.

Appears in: Unit 4: Contextual Applications, Unit 6: Integration and Accumulation