A tank’s temperature is modeled by T(t)=ig(3t^2-5t+1ig)^4; what is ?
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AP Calculus BC Quiz
Practice Chain Rule in AP Calculus BC with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.
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A tank’s temperature is modeled by T(t)=ig(3t^2-5t+1ig)^4; what is T′(t)?
This quiz focuses on Chain Rule, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Calculus BC.
Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.
A tank’s temperature is modeled by T(t)=ig(3t^2-5t+1ig)^4; what is T′(t)?
Explanation: This problem requires the chain rule to differentiate T(t)=(3t2−5t+1)4. The outer function is u4 and the inner function is u=3t2−5t+1. Using the chain rule, we differentiate the outer function to get 4u3, then multiply by the derivative of the inner function, which is 6t−5. This gives us T′(t)=4(3t2−5t+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.
If T(t)=ig(3t^2-5t+1ig)^4 models tank pressure, what is T′(t)?
Explanation: 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.
If g(x)=tan(πx2) models an angle, what is g′(x)?
Explanation: 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.
A cost model is C(x)=9x2+4x+1. What is C′(x)?
Explanation: 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.
A growth factor is G(x)=ln(x2+6x+10). What is G′(x)?
Explanation: 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.
A cost function is K(x)=(2x−1)7. What is K′(x)?
Explanation: 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.
A signal is modeled by S(x)=sin(4x3−x). What is S′(x)?
Explanation: 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.
A population model is P(t)=(1+t4)−3. What is P′(t)?
Explanation: 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.
A force is given by F(x)=ln(5x2−4x+1). What is F′(x)?
Explanation: 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.
A current is I(t)=tan(πt2). What is I′(t)?
Explanation: 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.
A model for pressure is P(t)=(cos(3t))5; what is P′(t)?
Explanation: This problem requires the chain rule to differentiate a composite function. The outer function is raising to the fifth power, while the inner function is cos(3t). To differentiate, first take the derivative of the outer function, which is 5 times the inner raised to the fourth power. Then, multiply by the derivative of the inner function, which is -sin(3t) times 3 or -3 sin(3t), resulting in -15 (cos(3t))⁴ sin(3t). A tempting distractor like choice A forgets the negative sign and the factor of 3 from the inner derivative. Always look for compositions where a function is plugged into another, such as a trigonometric function inside a power, to recognize when to apply the chain rule.
A sensor’s temperature is modeled by T(t)=ig(3t^2-5t+1\big)^4; what is T′(t)?
Explanation: This problem requires the chain rule to differentiate a composite function. The outer function is raising to the fourth power, while the inner function is the quadratic 3t² - 5t + 1. To differentiate, first take the derivative of the outer function, which is 4 times the inner function raised to the third power. Then, multiply by the derivative of the inner function, which is 6t - 5, resulting in 4(3t² - 5t + 1)³(6t - 5). A tempting distractor like choice A forgets to multiply by the inner derivative, leading to an incomplete answer. Always look for compositions where a function is plugged into another, such as a polynomial inside a power, to recognize when to apply the chain rule.
A medication level is M(t)=sin(2t3−t); what is M′(t)?
Explanation: This problem requires the chain rule to differentiate a composite function. The outer function is the sine function, while the inner function is the cubic 2t³ - t. To differentiate, first take the derivative of the outer function, which is cosine of the inner function. Then, multiply by the derivative of the inner function, which is 6t² - 1, yielding cos(2t³ - t)(6t² - 1). A tempting distractor like choice E swaps cosine for sine, which is the derivative of cosine instead of sine. Always look for compositions where a function is plugged into another, such as a polynomial inside a trigonometric function, to recognize when to apply the chain rule.
A model for pressure is P(x)=ig( rac{x^2+1}{x}ig)^5; what is P′(x)?
Explanation: To differentiate P(x)=(xx2+1)5, we use the chain rule. The outer function is u5 with derivative 5u4, and the inner function is u=xx2+1=x+x1 with derivative 1−x21. Applying the chain rule gives P′(x)=5(xx2+1)4(1−x21). Choice B has the wrong sign in the inner derivative. When the inner function is a quotient, simplify it first if possible, then differentiate—this often reveals a cleaner derivative.
A light sensor reads S(x)=cos(πx2); what is S′(x)?
Explanation: To differentiate S(x)=cos(πx2), we use the chain rule. The outer function is cos(u) with derivative −sin(u), and the inner function is u=πx2 with derivative 2πx. Applying the chain rule gives S′(x)=−sin(πx2)⋅2πx=−2πxsin(πx2). Choice A gives only the derivative of cosine without applying the full chain rule. When constants like π appear inside composite functions, treat them as part of the inner function and include them in the inner derivative.
A moving object’s speed is v(t)=t4+2t; what is v′(t)?
Explanation: For v(t)=t4+2t=(t4+2t)1/2, we apply the chain rule. The outer function is u1/2 with derivative 21u−1/2=2u1, and the inner function is u=t4+2t with derivative 4t3+2. The chain rule gives v′(t)=2t4+2t1⋅(4t3+2)=2t4+2t4t3+2. Choice B omits the factor of 21 from the square root derivative. When differentiating f(x), remember it equals 2f(x)f′(x).
For f(x)=7−4x1, what is f′(x)?
Explanation: This problem requires the chain rule to differentiate a function with a square root in the denominator. The function f(x) = 1/√(7 - 4x) can be rewritten as f(x) = (7 - 4x)^(-1/2), with outer function g(u) = u^(-1/2) and inner function u = 7 - 4x. By the chain rule, f'(x) = g'(u) · u'(x) = (-1/2)u^(-3/2) · (-4) = 2(7 - 4x)^(-3/2) = 2/(7 - 4x)^(3/2). Answer C incorrectly keeps the negative sign, giving -2/(7 - 4x)^(3/2), when the two negatives should multiply to give a positive result. When differentiating functions with radicals or fractional exponents, rewrite them with exponent notation to clearly apply the chain rule.
A growth model is G(x)=e2x2−3x. What is G′(x)?
Explanation: This problem requires the chain rule to differentiate G(x) = e^(2x² - 3x). The outer function is e^u and the inner function is u = 2x² - 3x. Applying the chain rule, we differentiate the outer function to get e^u (since the derivative of e^u is itself), then multiply by the derivative of the inner function, which is 4x - 3. Therefore, G'(x) = e^(2x² - 3x) · (4x - 3) = (4x - 3)e^(2x² - 3x). Choice A incorrectly provides just e^(2x² - 3x), omitting the crucial multiplication by the derivative of the exponent. Remember that when differentiating e^(f(x)), the result is always e^(f(x)) · f'(x).
A wave height is H(x)=sinx+2; what is H′(x)?
Explanation: 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.
A tank’s temperature is modeled by T(t)=ig(3t^2-5t+4ig)^7. What is T′(t)?
Explanation: 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.