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

Question 1: Chain rule derivatives (2 points)

1. Find \(\frac{d}{dx}\cos(3x^{2})\). (1 point)
2. Find \(\frac{d}{dx}e^{\sin x}\). (1 point)

Question 2: Composites from a table (3 points)

The table gives values of \(f\), \(g\) and their derivatives.

1. Let \(h(x)=f(g(x))\). Find \(h^{\prime}(1)\). (1 point)
2. Let \(k(x)=g(f(x))\). Find \(k^{\prime}(2)\). (1 point)
3. Let \(p(x)=f(x)\,g(x)\). Find \(p^{\prime}(3)\). (1 point)

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

The curve \(x^{3}+y^{3}=6xy\) is shown.

1. Find \(\frac{dy}{dx}\). (2 points)
2. Write an equation of the tangent line at \((3,3)\). (1 point)

Question 4: Derivative of an inverse (3 points)

Let \(f(x)=x^{5}+x^{3}+x\) and let \(g\) be its inverse function.

1. Show that \(f\) has an inverse function. (1 point)
2. Find \(g^{\prime}(3)\). (2 points)

Question 5: Inverse trig derivatives (2 points)

1. Find \(\frac{d}{dx}\tan^{-1}(x^{2})\). (1 point)
2. Find \(\frac{d}{dx}\sin^{-1}(3x)\). (1 point)

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

Let \(f(x)=xe^{-x}\).

1. Find \(f^{\prime\prime}(x)\). (1 point)
2. Find all \(x\) for which \(f^{\prime\prime}(x)=0\). (1 point)

Question 7: Second derivative, implicitly (2 points)

The curve \(xy=12\) passes through \((3,4)\).

1. Find \(\frac{dy}{dx}\) at \((3,4)\). (1 point)
2. Find \(\frac{d^{2}y}{dx^{2}}\) at \((3,4)\). (1 point)

Question 8: Bacteria growth rate (3 points)

A culture contains \(B(t)=2000e^{0.5\sqrt{t}}\) bacteria after \(t\) hours, for \(t>0\).

1. Find \(B^{\prime}(t)\). (2 points)
2. Find \(B^{\prime}(4)\) and state its units. (1 point)

See every chapter on the AP Calculus AB math page.