Tutorial Week 7
This tutorial will be about implicit differentiation and logarithmitic differentiation, as well as some application questions.
Implicit Differentiation
Q1: Given \(y\) and \(x\) are implicitly related, find \(\frac{dy}{dx}\) for \(3x^2y + 4y = 3y^4 + x\).
Q2: Which points on the curve \(xy = 3y + x\) have a tangent line perpendicular to \(y = 3x + 1\)?
Q3: What is the equation of the line tangent to \(\tan(x+y)=4x+1\) at \((0, \frac{\pi}{4})\)?
Q4: Find the derivative of \(x^2y^3 = \sqrt{xy + x} + 2y + 3\) with respect to \(t\) if \(x\) and \(y\) are functions of \(t\). What would the derivative be if \(x = t^3\) and \(y = 3t + 2\)?
Exponential Derivatives
Q5: Find \(f'(x)\) if \(f(x) = ln(ln(e^{e^{3x^3 + cos(x)}}))\).
Inverse Trigonometric Derivatives
Q6: Find the derivative of \(f(x) = 3^{cos^{-1}(x^2)} + tan^{-1}(\sqrt[5]{3x^3 + 1})\).
Logarithmitic Differentiation
Q7: Find the derivative of \(f(x) = x^x\).
Q8: Find the derivative of \(f(x) = \frac{\sqrt[3]{3x^4 + 4x}\sqrt{3cos(x) + 4}}{6x^3 + 2tan(x)}\).
Related Rates
Q9: A cylinder with a radius of \(4m\) is being filled at a rate of \(2m^3/min\). How quickly is the height of the water changing?
Q10: Ship A is located \(100km\) west of ship B. Ship A is travelling at \(20km/h\) east and ship B is travelling at \(30km/h\) north. At what rate does the distance between the two ships change after \(6\) hours?
Inverse Function Derivatives
Q11: Given \(f(x) = x^3 + ln(7x + 2) + 1\), find \((f^{-1})'(1)\).
