Chain Rule, Implicit Differentiation and Inverse Functions: AP Calculus AB Math Test Answer Key – PDF Download

AP Calculus Chain Test Answer Key: questions 1–5

Question 1: Chain rule derivatives (2 points)

1. \(-6x\sin(3x^{2})\)
2. \(\cos x\,e^{\sin x}\)

Scoring: 1 point for each correct derivative, including the inner derivative factor.

Question 2: Composites from a table (3 points)

1. \(h^{\prime}(1)=f^{\prime}(g(1))\,g^{\prime}(1)=f^{\prime}(3)\,g^{\prime}(1)=3(-1)\) = \(-3\)
2. \(k^{\prime}(2)=g^{\prime}(f(2))\,f^{\prime}(2)=g^{\prime}(4)\,f^{\prime}(2)=(-3)(-2)\) = \(6\)
3. \(p^{\prime}(3)=f^{\prime}(3)g(3)+f(3)g^{\prime}(3)=3(4)+1(2)\) = \(14\)

Scoring: 1 point per part. In (a) and (b), award the point only if the inner value is evaluated first.

Question 3: Implicit differentiation on a curve (3 points)

1. \(3x^{2}+3y^{2}\frac{dy}{dx}=6y+6x\frac{dy}{dx}\). Collect: \((3y^{2}-6x)\frac{dy}{dx}=6y-3x^{2}\). \(\frac{dy}{dx}=\frac{2y-x^{2}}{y^{2}-2x}\)
2. At \((3,3)\): \(\frac{6-9}{9-6}=-1\). \(y-3=-(x-3)\), or \(y=-x+6\).

Scoring: (a) 1 point for differentiating both sides correctly (product rule on \(6xy\), chain factor on \(y^{3}\)), 1 point for solving for \(\frac{dy}{dx}\). (b) 1 point for slope and equation.

Question 4: Derivative of an inverse (3 points)

1. \(f^{\prime}(x)=5x^{4}+3x^{2}+1\ge 1>0\) for all \(x\), so \(f\) is increasing and one-to-one. So f has an inverse.
2. \(f(1)=1+1+1=3\), so \(g(3)=1\). \(f^{\prime}(1)=5+3+1=9\). \(g^{\prime}(3)=\frac{1}{9}\)

Scoring: (a) 1 point for showing the derivative is positive for every x. (b) 1 point for finding f(1) = 3, 1 point for the reciprocal 1/9.

Question 5: Inverse trig derivatives (2 points)

1. \(\frac{2x}{1+x^{4}}\)
2. \(\frac{3}{\sqrt{1-9x^{2}}}\)

Scoring: 1 point each. No credit if the inner derivative factor is missing.

AP Calculus Chain Test Answer Key: questions 6–8

Question 6: Second derivative of a product (2 points)

1. \(f^{\prime}(x)=e^{-x}-xe^{-x}=e^{-x}(1-x)\); \(f^{\prime\prime}(x)=-e^{-x}(1-x)-e^{-x}=e^{-x}(x-2)\). \(f^{\prime\prime}(x)=(x-2)e^{-x}\)
2. \(e^{-x}>0\) always, so \(x=2\).

Scoring: (a) 1 point for the correct simplified second derivative. (b) 1 point for x = 2.

Question 7: Second derivative, implicitly (2 points)

1. \(y+x\frac{dy}{dx}=0\), so \(\frac{dy}{dx}=-\frac{y}{x}\). At \((3,4)\): \(-\frac{4}{3}\).
2. \(\frac{d^{2}y}{dx^{2}}=-\frac{x\frac{dy}{dx}-y}{x^{2}}=-\frac{-y-y}{x^{2}}=\frac{2y}{x^{2}}\). At \((3,4)\): \(\frac{8}{9}\).

Scoring: 1 point for each part. Check: \(y=12x^{-1}\) gives \(y^{\prime\prime}=24x^{-3}=\frac{24}{27}=\frac{8}{9}\).

Question 8: Bacteria growth rate (3 points)

1. \(B^{\prime}(t)=2000e^{0.5\sqrt{t}}\cdot 0.5\cdot\frac{1}{2\sqrt{t}}\) = \(\frac{500e^{0.5\sqrt{t}}}{\sqrt{t}}\)
2. \(B^{\prime}(4)=\frac{500e^{1}}{2}=250e\approx 679.6\). About 680 bacteria per hour at \(t=4\).

Scoring: (a) 1 point for the chain on the exponent, 1 point for the derivative of the square root. (b) 1 point for the value with units.

Scoring summary

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AP Calculus AB Chain Rule, Implicit Differentiation and Inverse Functions test answer key

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