Unit 5 - Calculus || Topic – Derivatives: Product, Quotient & Chain Rule Practice Set || IBDP AAHL, AASL, AIHL, AISL

IB Mathematics AA HL – Derivatives Practice Set

Topic: Product Rule, Quotient Rule, Chain Rule

Level: Higher Level (AA HL)

Total Questions: 14

Section A: Short Questions (Fundamental Applications of Rules)

Apply the appropriate differentiation rule: product, quotient, or chain rule. Give your answers in simplified form.

  1. Differentiate the function:

    $ f(x) = (3x^2 + 5)(\sin x) $
  2. Find $ \dfrac{dy}{dx} $ if

    $ y = x^2 e^x $
  3. Differentiate the function:

    $ f(x) = \ln x \cdot \tan x $
  4. Differentiate:

    $ f(x) = \dfrac{x^2 + 1}{\sqrt{x}} $
  5. Find the derivative of:

    $ y = \dfrac{\sin x}{x^2 + 1} $
  6. If $ f(x) = \dfrac{\ln x}{x^3} $, find $ f'(x) $
  7. Differentiate the following function:

    $ f(x) = \cos(3x^2 + 2) $
  8. Find $ \dfrac{dy}{dx} $ if

    $ y = \sqrt{1 + \tan x} $
  9. Differentiate:

    $ y = e^{(x^2 + 4x)} $
  10. Let $ f(x) = \ln(\sqrt{1 + e^x}) $. Find $ f'(x) $

Section B: Extended Response Questions (IB-Style)

These problems require extended reasoning and clear, structured solutions. Use correct mathematical notation throughout.

  1. Let $ f(x) = \sin(x^2 \cdot e^x) $

    (a) Find $ f'(x) $.
    (b) Comment on the effect of the chain rule in this derivative.
  2. The curve is defined implicitly by the equation:

    $ x^2 y + y^3 = \tan x $

    Find $ \dfrac{dy}{dx} $ in terms of $ x $ and $ y $, using implicit differentiation.
  3. Let $ f(x) = \dfrac{\ln(\sin x)}{x^2 + 1} $, where $ 0 < x < \pi $.

    (a) Find $ f'(x) $.
    (b) Hence, find the equation of the tangent to the curve at $ x = \dfrac{\pi}{4} $
  4. A balloon rises vertically such that its height in meters at time $ t $ seconds is given by:

    $ h(t) = 50 \ln(2t + 1) $

    A camera is placed 100 meters horizontally from the point where the balloon was released. Let $ D(t) $ be the distance between the camera and the balloon.

    (a) Express $ D(t) $ in terms of $ t $.
    (b) Find $ \dfrac{dD}{dt} $ when $ t = 3 $, giving appropriate units.
    (c) Interpret your result in the context of the motion of the balloon.