What is the derivative of y = sin^3(4x) with respect to x?
Answer and explanation
Correct answer: B. 12 sin^2(4x) cos(4x)
- Step 1: Apply the generalized power rule and chain rule: y = [sin(4x)]^3.
- Step 2: dy/dx = 3 [sin(4x)]^2 d/dx [sin(4x)].
- Step 3: The derivative of sin(4x) is cos(4x) 4 = 4 cos(4x).
- Step 4: Combine terms: dy/dx = 3 sin^2(4x) [4 cos(4x)] = 12 sin^2(4x) * cos(4x).
Answer
12 sin^2(4x) cos(4x).
Why not the others: '3 sin^2(4x) cos(4x)' forgets the inner chain rule derivative of 4x (factor of 4); '12 cos^3(4x)' differentiates incorrectly without preserving the power rule structure; '4 sin^2(4x) * cos(4x)' drops the power exponent 3.
Practise more Calculus and Mathematical Analysis.
Free questions with an explanation for every one, and full timed General Secondary Certificate (Tawjihiya Kuwait) mock tests.
More Calculus and Mathematical Analysis questions
- Evaluate the trigonometric limit: lim (x -> 0) [sin(5x) / (3x)].
- Find the derivative of the rational function f(x) = (2x - 3) / (x + 4) for x != -4.
- What is the slope of the tangent line to the curve y = x^3 - 3x^2 + 2 at the point where x = 3?
- Given the function f(x) = 2x^3 - 9x^2 + 12x - 5, find the x-coordinates of all local extrema (critical points).
- Determine the point of inflection for the cubic curve f(x) = x^3 - 6x^2 + 9x + 1.
- Evaluate the indefinite integral: integral of (6x^2 - 4x + 5) dx.