AP Calculus AB and BC

Derivative vs Differential

Q: What is the difference between dy and \Delta y?

\Delta y is the exact change in the function; dy is the change along the tangent line. The gap between them shrinks as dx shrinks.

Q: When is a differential better than an exact calculation?

When the exact value is awkward. Estimating \sqrt{16.2} from \sqrt{16} takes one multiplication: dy = \frac{1}{8} \cdot 0.2 = 0.025, giving 4.025.

The derivative is a rate: a function whose value at x is the instantaneous slope. The differential is an amount: the estimated change in y caused by a small step dx, found by multiplying the derivative by that step. Use the derivative when the question asks how fast, the differential when it asks how much.

Derivative

Use when: The question asks for a slope, a rate, or a function you will go on to set equal to zero, differentiate again, or evaluate.

Differential

Use when: The question asks how much a quantity changes, asks you to approximate a value, or asks how a measurement error propagates.

Side by side

	Derivative	Differential
What it is	A function, $f'(x)$	A quantity, $dy = f'(x)\,dx$
Units	Output units per input unit	Output units
Question it answers	How fast is $y$ changing?	How much does $y$ change?
Shows up in	Motion, optimization, curve sketching	Linear approximation, error estimation, separable equations
Common trap	Reporting a rate when the answer wanted an amount	Dropping the factor $dx$ , which leaves the estimate off by that scale

One multiplication separates them, and it changes what kind of object you are holding. The derivative $f'(x)$ is a function carrying units of output per input. Multiply it by a small step $dx$ and you get $dy$ , a single quantity in the units of the output, which is why $dy$ can be compared directly against a change in $y$ .

dy = f'(x)\,dx \qquad \Delta y = f(x + \Delta x) - f(x)

Here $dy$ is the estimate and $\Delta y$ is the truth. The tangent line is the object that trades one for the other: $L(x) = f(a) + f'(a)(x - a)$ says take the true change along the curve and replace it with the change along the tangent, an approximation that improves as the step shrinks.

Which way the estimate errs

Concavity decides the direction of the error. Where $f$ is concave up the tangent line sits below the curve, so $dy$ underestimates $\Delta y$ ; where $f$ is concave down the tangent sits above and $dy$ overestimates. Exam questions want that reason stated, not just the number.

Frequently asked questions

What is the difference between $dy$ and $\Delta y$ ?

$\Delta y$ is the exact change in the function; $dy$ is the change along the tangent line. The gap between them shrinks as $dx$ shrinks.

Is $\frac{dy}{dx}$ a fraction?

It is defined as a limit, not a quotient, but the differential form $dy = f'(x)\,dx$ lets you treat it like one. That is what licenses the separation step in a separable differential equation, and the chain rule is what justifies it.

When is a differential better than an exact calculation?

When the exact value is awkward. Estimating $\sqrt{16.2}$ from $\sqrt{16}$ takes one multiplication: $dy = \frac{1}{8} \cdot 0.2 = 0.025$ , giving $4.025$ .

In the CED: Unit 2: Defining the Derivative, Unit 4: Contextual Applications