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.
- Differentiate the function:
\( f(x) = (3x^2 + 5)(\sin x) \) - Find \( \dfrac{dy}{dx} \) if
\( y = x^2 e^x \) - Differentiate the function:
\( f(x) = \ln x \cdot \tan x \) - Differentiate:
\( f(x) = \dfrac{x^2 + 1}{\sqrt{x}} \) - Find the derivative of:
\( y = \dfrac{\sin x}{x^2 + 1} \) - If \( f(x) = \dfrac{\ln x}{x^3} \), find \( f'(x) \)
- Differentiate the following function:
\( f(x) = \cos(3x^2 + 2) \) - Find \( \dfrac{dy}{dx} \) if
\( y = \sqrt{1 + \tan x} \) - Differentiate:
\( y = e^{(x^2 + 4x)} \) - 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.
-
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. -
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. -
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 \( x = \dfrac{\pi}{4} \) -
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.
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