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 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 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(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 G(x) = ln(x² + 6x + 10). The chain rule states that when differentiating a composite function, we multiply the derivative of the outer function by the derivative of the inner function. The outer function is ln(u) with derivative 1/u, and the inner function is u = x² + 6x + 10 with derivative 2x + 6. Applying the chain rule: G'(x) = (1/(x² + 6x + 10)) · (2x + 6) = (2x + 6)/(x² + 6x + 10). Choice A incorrectly gives 1/(x² + 6x + 10), omitting the derivative of the inner function. Remember that d/dx[ln(f(x))] = f'(x)/f(x), which is a direct application of the chain rule to logarithmic functions.

This problem requires the chain rule to differentiate a composite function. The outer function is raising to the 7th power, and the inner function is 2x - 1. To differentiate, take the derivative of the outer function, which is 7 times the inner function raised to the 6th power. Then multiply by the derivative of the inner function, which is 2, resulting in 7(2x - $1)^6$ * 2 = 14(2x - $1)^6$. A tempting distractor is choice A, which forgets to multiply by the derivative of the inner function, which is 2. To recognize when to use the chain rule, look for compositions where one function is substituted into another, such as a linear expression inside a power function.

This problem requires the chain rule to differentiate a composite function. The outer function is the sine function, and the inner function is 4x³ - x. To differentiate, take the derivative of the outer function, which is cosine evaluated at the inner function. Then multiply by the derivative of the inner function, which is 12x² - 1, resulting in cos(4x³ - x) * (12x² - 1). A tempting distractor is choice B, which incorrectly uses sine instead of cosine for the outer derivative. To recognize when to use the chain rule, look for compositions where one function is substituted into another, such as a polynomial inside a trigonometric function.

This problem requires the chain rule to differentiate a composite function. The outer function is raising to the -3 power, and the inner function is 1 + t⁴. To differentiate, take the derivative of the outer function, which is -3 times the inner function raised to the -4 power. Then multiply by the derivative of the inner function, which is 4t³, resulting in -3(1 + $t⁴)^{-4}$ * 4t³ = -12t³ (1 + $t⁴)^{-4}$. A tempting distractor is choice C, which uses an exponent of -3 instead of lowering it to -4. To recognize when to use the chain rule, look for compositions where one function is substituted into another, such as a polynomial inside a negative power function.

This problem requires the chain rule to differentiate a composite function. The outer function is the natural logarithm, and the inner function is 5x² - 4x + 1. To differentiate, take the derivative of the outer function, which is 1 over the inner function. Then multiply by the derivative of the inner function, which is 10x - 4, resulting in (10x - 4) / (5x² - 4x + 1). A tempting distractor is choice A, which incorrectly places the derivative in the numerator without the logarithmic structure. To recognize when to use the chain rule, look for compositions where one function is substituted into another, such as a polynomial inside a logarithmic function.

This problem requires the chain rule to differentiate a composite function. The outer function is the tangent function, and the inner function is π t². To differentiate, take the derivative of the outer function, which is sec² evaluated at the inner function. Then multiply by the derivative of the inner function, which is 2π t, resulting in sec²(π t²) * 2π t. A tempting distractor is choice D, which incorrectly uses tangent instead of sec² for the outer derivative. To recognize when to use the chain rule, look for compositions where one function is substituted into another, such as a polynomial inside a trigonometric function like tangent.