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Calculus · u-substitution

I need to find

\int 2x \cos(x^2)\, dx

. Can you solve this for me?

I'm not going to solve it for you. But I'll help you work through it yourself. What do you notice about the relationship between

2x

and

x^2

Hmm,

2x

is the derivative of

x^2

. So maybe u-substitution?

The problem

You already know how this goes. You screenshot a problem, paste it into ChatGPT, and get the answer in three seconds. You copy it down. Assignment done. Full marks.

Then the midterm comes and you can't solve a single question. Because you never actually learned it. You just learned how to copy.

Every AI tool on the market is built to give you the answer as fast as possible. None of them are built to make sure you understand.

The difference

ChatGPT answers. Derive teaches.

Derive doesn't solve homework for you. It teaches you to solve it yourself.

Any answer engine

You ask → it answers → you forget.

You paste the problem. It hands you the solution. Assignment done, full marks, and nothing sticks. The midterm doesn't care that your homework was perfect.

Derive

It asks → you think → you get it.

It meets you at what you already know and refuses to skip the steps. Slower than copying. That's the point. When you get the answer, it's yours.

If you want instant answers, ChatGPT is free. Derive is for students who want to actually understand.

Why it works

What Derive does differently

Built on your actual course

Upload lecture notes, syllabi, and assignment sheets. Derive teaches with your professor's notation, terminology, and conventions. Not generic textbook explanations.

Catches the missing foundation

Trying to do derivatives but never really got the chain rule? Derive notices, backs up, and fills the gap before moving on.

You do the thinking

Derive asks what you already know, breaks the problem into steps, and explains a concept a different way when you're stuck.

When you finally arrive at the answer, you got there yourself. That's the difference between a grade and an education.

This is what learning with Derive looks like

I need to find

\int 2x \cos(x^2)\, dx

. Can you solve this for me?

I'm not going to solve it for you. But I'll help you work through it yourself. First: when you look at

2x \cos(x^2)

, does the structure remind you of any integration technique? What do you notice about the relationship between

2x

and

x^2

Hmm,

2x

is the derivative of

x^2

. So maybe u-substitution?

Exactly right. You spotted that

2x

is the derivative of

x^2

, which is the key insight. Now try it: if you let

u = x^2

, what does

du

equal, and what does the integral become in terms of

u

du = 2x\, dx

, so the integral becomes

\int \cos(u)\, du

... which is

\sin(u) + C

. So it's

\sin(x^2) + C

That's exactly right. And you worked through every step yourself. You recognized the chain rule pattern, chose the right substitution, and carried it through cleanly. That's u-substitution, and now you actually understand it.

Calculus

What you walk away with

The solution, found by you

Every step in that conversation came from the student. Derive just refused to skip any.

The weak spot, caught early

The exact place your understanding gets fuzzy shows up in the conversation. Better there than on the exam.

Notes from your mistakes

One click turns the session into study notes built from where you struggled. Not a summary you could have Googled.

Try it yourself

Getting started

How it works

Add your courses.

Create a workspace for each class. Upload your lecture notes, slides, and assignments.

Start a conversation.

Ask about a problem, a concept, anything you're stuck on. Derive guides you through it step by step.

Make it stick.

When you work through something and finally get it, hit “Generate Notes.” Derive creates personalized study notes based on what you specifically struggled with. Not generic summaries. Notes you'll actually use.

Start learning for free

Who this is for

Universities are designed for a very specific type of student. The kind who sits in a lecture, takes notes, reviews them once, and gets it. That works for maybe 5% of people.

This is for the other 95%.

If you're someone who genuinely wants to learn but lectures aren't enough. If you've ever thought “I'm just not smart enough for this” when the real problem was that nobody taught it in a way that clicked for you. If you're tired of choosing between copying AI answers and struggling alone.

Your notes, based on what you actually struggled with

Generated study notes

From your session on definite integrals with substitution

Key mistake: Forgetting that C cancels in definite integrals

When evaluating a definite integral, you don't need the + C. The constant of integration cancels when you subtract F(a) from F(b). You were adding it out of habit from indefinite integrals. That's a common trap.

Sign error during substitution

When you substituted $u = 1 - x$ , you forgot that $du = -dx$ , which flips the sign of the entire integral. You caught it on the second pass. Next time, write out $du$ explicitly before substituting to avoid this.

Takeaway: Always check substitution with negative values

When your limits of integration involve negative numbers or when the substitution introduces a negative, pause and verify the sign. Plug the original limits into u to get the new limits. Don't guess.

Calculus

Not generic summaries. Notes built from your specific mistakes and breakthroughs.

Pricing

A private tutor is $40 to $80 an hour. Derive Pro is $20 a month.

Free

10 messages per day. No credit card. Full access to the platform. See for yourself how it works.

AI-guided tutoring
Course material upload
Personalized study notes

Derive Pro

$20/month

2,000 messages per month. Enough to work through full problem sets and study for exams without rationing questions.

2,000 messages per month
Everything in Free
All future features included

Cancel anytime in your account settings.

Common questions

Can't I just trick it into giving me the answer?

Probably, if you try hard enough. But if you wanted answers handed to you, ChatGPT is free. Derive is for the exam, not just the assignment.

Is the free plan actually free?

10 messages a day, every day. Full access to tutoring, course uploads, and study notes. No credit card.

What can I upload?

Lecture notes, syllabi, assignment sheets, problem sets. PDFs and text files work best, up to 10MB each. Derive reads them and teaches using your course's notation and terminology.

Which subjects does it handle?

Anything you're taking. It's built for the courses where students get stuck most: calculus, physics, statistics, chemistry, computer science. Equations render properly both ways, so the math looks like your textbook, not code.

What happens if I hit the Pro limit?

2,000 messages a month is around 65 a day, every day. You'll see a heads-up at 1,000, and the count resets on your billing date.

Can I cancel Pro?

Anytime, in your account settings.

Derive turns “I copied the answer” into “I figured it out myself.” If that's the student you want to be, this is built for you.

Got a problem set due this week? Bring it.

Start learning for free

No credit card required. Takes 30 seconds.