5.9 Differentiation rules and related ratesIB Maths: Applications and Interpretation HL: Flashcards

What these 13 flashcards ask

Derivative of x^n for any rational n?
\dfrac{d}{dx}\sin x and \dfrac{d}{dx}\cos x?
\dfrac{d}{dx}\tan x?
\dfrac{d}{dx}e^x and \dfrac{d}{dx}\ln x?
State the chain rule.
State the product rule.
State the quotient rule.
Differentiate \sin(2x+5).
Differentiate \ln(3x+2).
Differentiate \sqrt{x}.
How do you solve a related-rates problem?
How do you find the maximum of a model?
Which angle mode must the GDC use for derivatives of trig functions?

Exam questions on 5.9 Differentiation rules and related rates

The depth of water, $h$ metres, in a harbour $t$ hours after midnight is modelled by $h(t)=6+2\sin(0.5t)$ for $0\le t\le 12$ , where the angle is in radians.
Find the first time after midnight at which the depth is greatest, and state the greatest depth.2 marks
The concentration of a drug in a patient's blood, $C$ mg l $^{-1}$ , is modelled by $C(t)=\dfrac{20t}{t^2+4}$ for $t\ge0$ , where $t$ is the time in hours after the drug is given.
Find the rate of change of the concentration at $t=1$ and state whether the concentration is increasing or decreasing.2 marks
The concentration of a chemical in a reaction vessel, $g$ mol dm $^{-3}$ , is modelled by $g(t)=\sqrt{t}\,e^{-0.5t}$ for $t>0$ , where $t$ is the time in minutes after mixing.
Find an expression for $\dfrac{dg}{dt}$ .3 marks

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).