Implicit Differentiation & Related Rates
IB Mathematics AA · HL · Shadow Worksheet
Practice
Question 1
Find \(\dfrac{dy}{dx}\) for each of the following:
(a)\(x^2+y^2=25\)
(b)\(x^3+y^3=6xy\)
(c)\(\sin(xy)=x+y\)
(d)\(e^y+x^2y=3\)
(a)
Differentiate implicitly
\(2x+2y\dfrac{dy}{dx}=0\)
Answer
\(\dfrac{dy}{dx}=-\dfrac{x}{y}\)
(b)
Differentiate
\(3x^2+3y^2\dfrac{dy}{dx}=6y+6x\dfrac{dy}{dx}\Rightarrow\dfrac{dy}{dx}(3y^2-6x)=6y-3x^2\)
Answer
\(\dfrac{dy}{dx}=\dfrac{2y-x^2}{y^2-2x}\)
(c)
Chain rule on LHS
\(\cos(xy)\!\left(y+x\dfrac{dy}{dx}
ight)=1+\dfrac{dy}{dx}\Rightarrow\dfrac{dy}{dx}[x\cos(xy)-1]=1-y\cos(xy)\)
Answer
\(\dfrac{dy}{dx}=\dfrac{1-y\cos(xy)}{x\cos(xy)-1}\)
(d)
Differentiate
\(e^y\dfrac{dy}{dx}+2xy+x^2\dfrac{dy}{dx}=0\Rightarrow\dfrac{dy}{dx}(e^y+x^2)=-2xy\)
Answer
\(\dfrac{dy}{dx}=\dfrac{-2xy}{e^y+x^2}\)
Question 2
Find the equation of the tangent to \(x^2+xy+y^2=7\) at the point \((1,2)\).
Implicit differentiation
\(2x+y+x\dfrac{dy}{dx}+2y\dfrac{dy}{dx}=0\Rightarrow\dfrac{dy}{dx}=\dfrac{-2x-y}{x+2y}\)
At \((1,2)\)
\(\dfrac{dy}{dx}=\dfrac{-4}{5}\)
Answer
\(y-2=-\dfrac{4}{5}(x-1)\Rightarrow 5y+4x=14\)
Question 3
A spherical balloon is being inflated so that its volume increases at 50 cm³/s. Find the rate at which the radius is increasing when the radius is 5 cm.
Volume formula
\(V= frac{4}{3}\pi r^3\Rightarrow\dfrac{dV}{dt}=4\pi r^2\dfrac{dr}{dt}\)
Substitute
\(50=4\pi(25)\dfrac{dr}{dt}\Rightarrow\dfrac{dr}{dt}=\dfrac{50}{100\pi}\)
Answer
\(\dfrac{dr}{dt}=\dfrac{1}{2\pi}pprox0.159\) cm/s
Question 4
A 5 m ladder leans against a wall. The base slides away from the wall at 0.5 m/s. Find the rate at which the top of the ladder slides down when the base is 3 m from the wall.
Pythagoras
\(x^2+y^2=25\Rightarrow 2x\dfrac{dx}{dt}+2y\dfrac{dy}{dt}=0\)
At \(x=3\)
\(y=4;\;2(3)(0.5)+2(4)\dfrac{dy}{dt}=0\Rightarrow\dfrac{dy}{dt}=-\dfrac{3}{8}\)
Answer
Top slides down at \(\dfrac{3}{8}=0.375\) m/s
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