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find dy/dx for y = sin(3x²)
Answer
dy/dx = 6x·cos(3x²)
Step 1 — Spot the composition
y = sin(u) with u = 3x² — an outer function wrapped around an inner one, so the chain rule applies.
Step 2 — Differentiate the outer function
d/du [sin(u)] = cos(u), so we get cos(3x²), keeping the inside untouched.
Step 3 — Multiply by the inner derivative
du/dx = 6x, so dy/dx = cos(3x²) · 6x = 6x·cos(3x²).
Why this works
The chain rule composes rates of change: how fast sin changes with respect to u, times how fast u changes with respect to x.
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