Express the relationship between a small change in x and the corresponding change in y in the form .

y = sin(3x2)

Question

Express the relationship between a small change in x and the corresponding change in y in the form .

y = sin(3x2)

BlackTom AI · Accepted Answer

To relate a small change in x to a change in y for y = sin(3x^2), I start by applying the chain rule. The derivative dy/dx is the derivative of sin(u) with respect to x, where u = 3x^2. First, differentiate the outer function: d/dx [sin(u)] = cos(u) * du/dx. Next, differentiate the inner function: du/dx = d/dx [3x^2] = 6x. Multiplying these together gives dy/dx = cos(3x^2) * 6x = 6x cos(3x^2).

Now, the differential dy is related to dx by dy = (dy/dx) dx. Substituting the expression we found for dy/dx yields dy = 6x cos(3x^2) dx.

Evaluating the remaining options:
- Option 1: dy = 6x cos(3x^2) dx — this matches the result from the chain rule and the differential relation, so it is correct.
- Option 2: dy = -6x cos(3x^2) dx — the negative sign would imply the opposite slope, which is incorrect given dy/dx is positive 6x cos(3x^2).
- Option 3: dy = -6 cos(3x^2) dx — missing the factor x from the inner derivative, and the sign is incorrect.
- Option 4: dy = 6 cos(3x^2) dx — missing the x factor, which is essential from the derivative of 3x^2, so this is incorrect.

Express the relationship between a small change in x and the corresponding change in y in the form . y = sin(3x2) 单项选择题

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