Find the derivative dy/dx of the function y = (3x^2 + 1)^4.
Answer and explanation
Correct answer: D. dy/dx = 24x(3x^2 + 1)^3
- Step 1: Apply the Chain Rule: d/dx [u(x)^n] = n u(x)^(n - 1) u'(x).
- Step 2: Here, u(x) = 3x^2 + 1, and n = 4.
- Step 3: The derivative of the inner function u'(x) = d/dx(3x^2 + 1) = 6x.
- Step 4: Combine terms: dy/dx = 4 (3x^2 + 1)^3 (6x) = 24x(3x^2 + 1)^3.
Answer
dy/dx = 24x(3x^2 + 1)^3.
Why not the others: '4(3x^2 + 1)^3' forgets the chain rule inner derivative u'(x); '12x(3x^2 + 1)^3' calculates 2 6x instead of 4 6x; '6x(3x^2 + 1)^3' forgets the outer power coefficient 4.
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