AP Calculus AB and BC

Derivative Practice Checker: Any Correct Form Passes

The Derivative Practice Checker shows a function, you type its derivative, and it grades by numeric equivalence, not string matching: any algebraically equal form is accepted and a constant-offset slip is always caught. A miss gets a hint naming the rule to reconsider, never the answer.

Problem

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Differentiate

f(x) = x^3 - 5x + 2

The tool shows one function at a time and asks you to type its derivative into the box. Instead of comparing your typing to one stored string, it grades by numeric equivalence. Writing $g$ for the expression you type and $f'$ for the reference derivative, it evaluates both at roughly a dozen points drawn from inside the shared domain and marks you correct only when they agree at every one of them.

\left| g(x_i) - f'(x_i) \right| < 10^{-6} \cdot \max\left(1, \left| f'(x_i) \right|\right) \quad \text{at every sampled } x_i

Any correct form passes

Because the check is numeric and not textual, every algebraically equal answer counts. For $\tan x$ the tool accepts both $\sec^2 x$ and $1 + \tan^2 x$ ; for $\cos^2 x$ it accepts both $-2\sin x\cos x$ and $-\sin(2x)$ . A quiz that stored a single answer string would reject the others, even though they are the same function. This tool cares only about the function you wrote, not how you wrote it.

Sampling at many points, not one, is what makes that trust safe. An answer that is off by a constant, a dropped $+1$ say, disagrees at every point and is caught. So is a chain-rule slip that forgets the inner factor, because it changes the shape of the function, not just its value at one place. A single-point check could be fooled by either; a dozen-point check is not.

An empty box returns a gentle prompt to type something. It is never marked wrong.
Input the parser cannot read returns the parser's own message, and you are not penalized, because a typo is not a math mistake.
A parseable answer that disagrees somewhere is marked not equivalent yet, with a hint toward the rule to reconsider.
A parseable answer that agrees at every sampled point is marked correct.

The ten problems are not grouped under rule headings. Before you can differentiate anything you have to read the function and decide which rule governs it, which is a graded skill in its own right: the CED calls it selecting procedures for calculating derivatives (Topic 3.5). Reading the structure first is the habit the tool is built to reward.

When the function looks like	Reach for	CED topic
A sum of power terms, $x^3 - 5x + 2$	Power rule, term by term	Topics 2.5 and 2.6
Two functions multiplied, $x^2\sin x$	Product rule	Topic 2.8
One expression over another, $\frac{2x+1}{x^2+1}$	Quotient rule	Topic 2.9
A function inside a function, $\sin(3x^2)$	Chain rule	Topic 3.1
A trig function on its own, $\tan x$	A memorized trig derivative	Topic 2.10
An exponential, $e^{2x}$	Exponential rule with the chain rule	Topics 2.7 and 3.1
A logarithm as a factor, $x\ln x$	The $\ln$ derivative plus the product rule	Topics 2.7 and 2.8

These rules combine, and the tool leans on that. $x^2 e^x$ is a product of two functions you can each differentiate; $\sqrt{x^2+1}$ is a power wrapped around $x^2+1$ , so it is a chain; $\cos^2 x$ means $(\cos x)^2$ , a chain whose outer operation is squaring. Name the outermost structure first, then work inward, and the rule to start with becomes clear.

Take the function $f(x) = \cos^2 x$ . Read its structure before reaching for a rule: $\cos^2 x$ means $(\cos x)^2$ , a function raised to a power, so the outer operation is squaring and the inner function is $\cos x$ . That makes this a chain rule problem (Topic 3.1), not a product.

f(x) = (\cos x)^2, \qquad \text{outer } u^2, \quad \text{inner } u = \cos x

Differentiate the outer square, leaving the inside untouched, then multiply by the derivative of the inside:

f'(x) = 2\cos x \cdot (-\sin x) = -2\sin x\cos x

Now use the double-angle identity $\sin(2x) = 2\sin x\cos x$ . The same derivative can be written $-\sin(2x)$ , and both forms are exactly the same function:

-2\sin x\cos x = -\sin(2x)

Type either one into the checker and it passes, because it evaluates your expression at a dozen points and both forms return identical values everywhere. Reveal the derivation afterward and the tool shows the standard form, $f'(x) = -2\sin x\cos x$ , alongside the rule-by-rule steps.

How this tool teaches instead of solving

You produce the derivative and type it in; a solver would produce it for you. The tool never shows the answer until you have committed to one or asked to see the derivation. Grading is by numeric equivalence, so any algebraically correct form is accepted; a string-matching quiz would reject a right answer written a different way. A wrong answer returns a rule-scoped hint that names the rule to reconsider but never states the derivative, so the retry is your own reasoning. You decide which differentiation rule applies before differentiating, the skill graded in Topic 3.5, rather than having the method chosen for you. Because it samples about a dozen points, a constant-offset or dropped-factor mistake is always caught, unlike a check at a single point. The full rule-by-rule derivation and the standard-form answer stay behind a reveal, available only after you try, so you check your method instead of copying it.

Frequently asked questions

Does the checker accept a derivative written in a different form?

Yes. It grades by numeric equivalence, not by matching your text to a stored answer, so any algebraically equal form counts. It evaluates both your expression and the reference at about a dozen points and marks you correct when they agree everywhere. For $\tan x$ both $\sec^2 x$ and $1 + \tan^2 x$ pass; for $\cos^2 x$ both $-2\sin x\cos x$ and $-\sin(2x)$ pass. Write the derivative in whatever form your work produced.

I typed a correct derivative but it said it could not read the expression. Why?

That is a grammar problem, not a wrong answer, and it is not counted against you. The parser needs functions in parentheses, as in $\sin(x)$ , $\cos(x)$ , or $\ln(x)$ ; powers written with a caret, as in x^2; and the exponential written as e^(2x) or exp(2x). One more quirk: log means natural log here. Fix the notation and check again, and the tool grades the corrected expression.

Will it catch a derivative that is off by a constant?

Yes. Because the tool samples about a dozen points rather than one, an answer that differs from the reference by a constant, such as a dropped $+1$ , disagrees at every point and is marked not equivalent. The same is true of a chain-rule slip that omits the inner factor, since that changes the function's shape. A single-point check could accidentally pass either mistake; this one does not.

Does the tool show the steps?

Only after you have committed to an answer or asked for them. On a miss you first get a hint that names the rule to reconsider without stating the derivative, so your next attempt is still your own work. You can then reveal the full rule-by-rule derivation and the standard-form answer, which are the only place the tool ever writes out $f'(x)$ . There is no score and no timer; the progress dots simply fill in as you solve each problem.

Which differentiation rules does the practice cover?

The ten problems span the core AB rules: the power rule, product rule, quotient rule, and chain rule, plus the derivatives of trigonometric, exponential, and logarithmic functions. These live in Units 2 and 3 of the CED and are foundational for both AP Calculus AB and BC, so BC students use the same tool to keep the mechanics sharp.