Differentiate f(x) = sin(3x^2) with respect to x.

Differentiate f(x) = sin(3x^2) with respect to x.

• A. 6x cos(3x^2)
• B. cos(3x^2)
• C. 3x^2 cos(3x^2)
• D. -6x sin(3x^2)

Answer: (A)

Chain rule applied to a composite function: differentiate the outer sine and multiply by the derivative of the inner function. Here the inner function is 3x^2, whose derivative is 6x, so the derivative becomes cos(3x^2) times 6x, i.e., 6x cos(3x^2). The other expressions miss this inner-derivative factor or mix up sine and cosine: dropping the 6x factor gives cos(3x^2); treating the inner function incorrectly gives 3x^2 cos(3x^2); and a negative sine term would come from differentiating cosine, not sine.

Chain rule applied to a composite function: differentiate the outer sine and multiply by the derivative of the inner function. Here the inner function is 3x^2, whose derivative is 6x, so the derivative becomes cos(3x^2) times 6x, i.e., 6x cos(3x^2). The other expressions miss this inner-derivative factor or mix up sine and cosine: dropping the 6x factor gives cos(3x^2); treating the inner function incorrectly gives 3x^2 cos(3x^2); and a negative sine term would come from differentiating cosine, not sine.