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This is a pre-canned conversation, not a live chat — but every response is the kind of explanation PrismAI gives in pedagogical mode, with citations to the source textbook.
PrismAI · Pedagogical mode
Calculus 1 · Spring semester syllabus
You
Student
I have to find d/dx of sin(x²) for tomorrow's quiz. I keep getting cos(x²) but my friend says that's wrong. What am I missing?
P
PrismAI
You're missing the chain rule. cos(x²) is what you'd get if the inside were just x — but it isn't, it's x². When the inside of a function is itself a function, you have to differentiate the outside, then multiply by the derivative of the inside.
Let me walk you through it the way Stewart introduces it in Chapter 2.5:
Stewart, Calculus 8e— §2.5 — The Chain Rule, p. 198
P
PrismAI
Step 1. Identify the outside and inside.
• Outside: sin( · )
• Inside: x²
Step 2. Differentiate the outside, leaving the inside alone.
d/dx[sin(u)] = cos(u), so we get cos(x²).
Step 3. Multiply by the derivative of the inside.
d/dx[x²] = 2x.
Result: d/dx[sin(x²)] = cos(x²) · 2x = 2x · cos(x²).
So your friend was right — the missing factor is the 2x. That's what the chain rule gives you that the power rule alone doesn't.
You
Student
Why doesn't the power rule work here?
P
PrismAI
Because the power rule applies to expressions of the form xⁿ — a variable raised to a constant power. Here you have sin(x²): a *function of* x², not a power of x.
A useful test: ask whether the expression looks like 'something to the power of a number.' sin(x²) doesn't — it's a sine function whose argument happens to be a power. Different rule.
This distinction trips up most students on their first chain-rule problem, which is why Stewart spends most of §2.5 on it before moving on.
Stewart, Calculus 8e— §2.5 — Examples 1–4, p. 199
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