Identify outer = sin(u) and inner = 3x². Differentiate outer: cos(u) = cos(3x²). Differentiate inner: 6x. Multiply: 6x · cos(3x²). Option A forgets to multiply by the inner derivative. Option B applies the inner derivative inside the trig function — a common error of 'differentiating inside the argument' instead of multiplying outside. Option C has the wrong sign (sin differentiates to +cos, not -cos).
Question 2 True / False
The chain rule is mainly needed when a function is raised to a power, like (3x + 1)⁵.
TTrue
FFalse
Answer: False
The chain rule applies to any composite function — whenever one function is nested inside another. sin(x²), e^(3x), ln(x² + 1), and √(x + 1) all require the chain rule even though none of them is a polynomial raised to a power. The trigger is composition, not exponents specifically.
Question 3 Short Answer
Identify the outer and inner functions in h(x) = e^(x² + 1), then find h'(x).
The inner function is what gets evaluated first: x² + 1. The outer function is applied to that result: e raised to whatever the inner gives. By the chain rule, h'(x) = f'(g(x)) · g'(x) = e^(x²+1) · 2x. Note that the derivative of eᵘ is eᵘ — the exponential function is its own derivative — so only the inner derivative 2x is new.