This problem requires the chain rule to differentiate a composite function. The outer function is the square root, or one-half power, while the inner function is sin x + 2. To differentiate, first take the derivative of the outer function, which is one-half times the inner to the negative one-half power. Then, multiply by the derivative of the inner function, which is cos x, resulting in cos x / (2 √(sin x + 2)). A tempting distractor like choice C doubles the denominator by forgetting the one-half factor. Always look for compositions where a function is plugged into another, such as a trigonometric function inside a root, to recognize when to apply the chain rule.

This problem requires the chain rule to differentiate a composite function. The function T(t) = (3t² - 5t + 4)⁷ has an outer function f(u) = u⁷ and an inner function u = 3t² - 5t + 4. By the chain rule, T'(t) = f'(u) · u'(t) = 7u⁶ · (6t - 5) = 7(3t² - 5t + 4)⁶ · (6t - 5). A common error is forgetting to multiply by the derivative of the inner function, which would give answer A: 7(3t² - 5t + 4)⁶. To recognize when to use the chain rule, look for a function nested inside another function—here, a polynomial inside a power function.

This problem requires the chain rule to differentiate $T(t)=(3t^2-5t+1)^4$. The outer function is $u^4$ and the inner function is $u=3t^2-5t+1$. Using the chain rule, we differentiate the outer function to get $4u^3$, then multiply by the derivative of the inner function, which is $6t-5$. This gives us $T'(t)=4(3t^2-5t+1)^3(6t-5)$. Choice B incorrectly gives only the derivative of the inner function without applying the chain rule. When you see a composite function like $(expression)^n$, always identify the inner expression first, then multiply the power rule result by the inner derivative.

This problem requires the chain rule to differentiate a negative power function. The function R(x) = 1/(x³ - 2)² = (x³ - 2)^(-2) has an outer function f(u) = u^(-2) and an inner function u = x³ - 2. By the chain rule, R'(x) = -2u^(-3) · u'(x) = -2(x³ - 2)^(-3) · 3x² = -6x²/(x³ - 2)³. Option A shows -2/(x³ - 2)³, forgetting to include the derivative of the inner function (3x²). Remember that when differentiating $1/g(x)^n$, the result is -n·g'(x)/g(x)^(n+1), which helps verify proper chain rule application.

This problem requires the chain rule to differentiate a composite function. The function T(t) = (3t² - 5t + 1)⁴ has an outer function f(u) = u⁴ and an inner function u = 3t² - 5t + 1. By the chain rule, T'(t) = f'(u) · u'(t) = 4u³ · (6t - 5) = 4(3t² - 5t + 1)³ · (6t - 5). A common error would be to forget the inner derivative and choose option A, which only shows 4(3t² - 5t + 1)³. When applying the chain rule, always identify both the outer and inner functions, differentiate each separately, then multiply the results together.

This problem requires the chain rule to differentiate a negative power function. The function h(x) = (1 - 5x)^(-3) has an outer function f(u) = u^(-3) and an inner function u = 1 - 5x. By the chain rule, h'(x) = -3u^(-4) · u'(x) = -3(1 - 5x)^(-4) · (-5) = 15(1 - 5x)^(-4). Option C shows -15(1 - 5x)^(-4), incorrectly handling the double negative from (-3) × (-5). When applying the chain rule with negative exponents and negative derivatives, carefully track the sign changes to avoid errors.

This problem requires the chain rule to differentiate a composite trigonometric function. The function g(x) = tan(πx²) has an outer function f(u) = tan(u) and an inner function u = πx². By the chain rule, g'(x) = sec²(u) · u'(x) = sec²(πx²) · 2πx = 2πx sec²(πx²). Option B shows 2x sec²(πx²), forgetting to include the π factor from differentiating πx². When the inner function contains a constant multiplier, that constant must be included in the final derivative through the chain rule multiplication.

This problem requires the chain rule to differentiate c(t) = ((2t-1)/(t+3))⁵. The chain rule tells us to multiply the derivative of the outer function by the derivative of the inner function. The outer function is u⁵ with derivative 5u⁴, and the inner function is u = (2t-1)/(t+3), which requires the quotient rule: u' = [2(t+3) - (2t-1)(1)]/(t+3)² = 7/(t+3)². Applying the chain rule: c'(t) = 5((2t-1)/(t+3))⁴ · (7/(t+3)²). Choice B incorrectly uses 2/(t+3) as the derivative of the inner function, which appears to be just the derivative of the numerator divided by the denominator. When the inner function is a quotient, you must use the quotient rule to find its derivative before applying the chain rule.

This problem requires the chain rule to differentiate C(x) = √(9x² + 4x + 1) = (9x² + 4x + 1)^(1/2). The chain rule tells us to multiply the derivative of the outer function by the derivative of the inner function. The outer function is u^(1/2) with derivative (1/2)u^(-1/2), and the inner function is u = 9x² + 4x + 1 with derivative 18x + 4. Applying the chain rule: C'(x) = (1/2)(9x² + 4x + 1)^(-1/2) · (18x + 4) = (18x + 4)/(2√(9x² + 4x + 1)). Choice C incorrectly gives 1/(2√(9x² + 4x + 1)), forgetting to multiply by the derivative of the inner function (18x + 4). When differentiating square roots, remember to apply the chain rule: the derivative includes both the power rule for the outer function and the derivative of what's inside.

This problem requires the chain rule to differentiate T(x) = sin(5x³ - 2x). The chain rule tells us to multiply the derivative of the outer function (sine) by the derivative of the inner function. The outer function is sin(u) with derivative cos(u), and the inner function is u = 5x³ - 2x with derivative 15x² - 2. Applying the chain rule: T'(x) = cos(5x³ - 2x) · (15x² - 2). Choice A incorrectly gives only cos(5x³ - 2x), forgetting to multiply by the derivative of the inner function. To apply the chain rule correctly, always identify your inner and outer functions first, then multiply their derivatives in the proper order: outer derivative evaluated at inner, times inner derivative.