Calculus help for the steps you write by hand

alai is a notes app for iPad and the web with Ally, an adaptive AI tutor that follows your work and answers your questions out loud. A calculus solution is usually a few calculus steps with a lot of algebra between them, and a slip in either one carries through to the answer.

Where calculus goes wrong on paper

Paul Dawkins wrote a list of the errors that came up often in his Calculus I class at Lamar University. He points out that many of them aren't really calculus errors. They're slips that happen during a calculus problem, and mistakes in calculus notation (Common Math Errors: Calculus Errors).

Limits

Plugging in is the first thing to try, and getting 0/0 means there's more work to do. For the limit of (x² − 9)/(x − 3) as x → 3, plugging in 3 gives 0/0. Factoring the top as (x − 3)(x + 3) and canceling x − 3 leaves x + 3, so the limit is 6. Writing that 0/0 equals 0, or that the limit doesn't exist, stops one step too early.

Dawkins also flags dropped limit notation. Once you write lim on the first line, it belongs on every line until you take the limit. Without it, a line like (x² − 9)/(x − 3) = x + 3 = 6 says something that isn't true.

Derivatives

• The chain rule's last factor. The derivative of (3x² + 1)⁵ is 5(3x² + 1)⁴ · 6x, which simplifies to 30x(3x² + 1)⁴. Stopping at 5(3x² + 1)⁴ leaves out the derivative of the inside, 6x.
• Multiplying the derivatives in a product. The derivative of x² sin x is 2x sin x + x² cos x. Writing 2x cos x multiplies the two derivatives together, which isn't the product rule.
• The order in the quotient rule. The derivative of f/g is (f′g − fg′)/g². Swapping the two terms on top flips the sign of the whole answer.

Integrals

• The constant of integration. ∫ 2x dx = x² + C. Leaving off the + C drops every other antiderivative, such as x² + 5. Dawkins notes that this matters most in differential equations, where the constant appears partway through the solution.
• The power rule at n = −1. The rule ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C works for every n except −1, because it would divide by zero. ∫ x⁻¹ dx is ln|x| + C.
• Choosing u. For ∫ 2x cos(x²) dx, let u = x². Then du = 2x dx, and the integral becomes ∫ cos u du = sin u + C, which is sin(x²) + C. Choosing u = cos(x²) gives a du that doesn't appear anywhere in the integral.
• Bounds after a substitution. In a definite integral, the bounds are values of x. After you switch to u, either change the bounds to u-values or switch back to x before you plug them in.

How Ally helps as you work

For calculus, use hold to talk. Write your work by hand in Homework help, and when you're unsure about a step or stuck on what comes next, hold the talk button and ask. Ally answers out loud in a few sentences. The reply is meant to point you at the next piece and leave the step itself to you.

Say you've worked out that the derivative of (3x² + 1)⁵ is 5(3x² + 1)⁴. You hold the talk button, say what you got and ask whether you're done. Ally tells you the outside part is right and asks a question like “What's the derivative of the inside, 3x² + 1?” You work out 6x and multiply it in yourself.

Or say you're starting ∫ 2x cos(x²) dx and you ask which part to call u. Ally leaves the choice to you and asks a question like “Is there a piece inside the cosine whose derivative also shows up in the integral?”

If a topic is new to you, start in Learn, where Ally teaches it in a few short spoken steps on the board before you practice. You can also upload a photo or PDF of your problem set and write your work on the page.

Common calculus mistakes, one page each

• Forgetting the chain rule: the derivative of the inside function
• The product rule: why the derivative of a product isn't the product of the derivatives
• The constant of integration: why + C belongs on every indefinite integral
• 0/0 limits: what to do when plugging in gives 0/0

Two algebra pages are useful in calculus too: exponent rules, for derivatives of powers, and canceling terms in a fraction, for simplifying limits.

Comparing alai with other apps

If you're weighing alai against another study tool, see alai vs Khanmigo, alai vs ChatGPT study mode and alai vs Photomath. For checking any handwritten math yourself, see how to check handwritten math step by step.