Exercises

01

Implicit differentiation practice

Use implicit differentiation to find $y^{\prime }=\frac{dy}{dx}$ when given each of these relationships:

• (a) $x^{2/3}+y^{2/3}=4$
• (b) $x{e}^y=x-y$

02

Derivative of $\ln x$

Compute the derivatives of the following functions:

• (a) $f(x)=\ln \left({\sin }^{2}x\right)$
• (b) $g(x)=\ln |1+t-t^3|$

03

Logarithmic differentiation

Find the derivative of $f(x)=x^x$.

04

Related rates - inflating sphere

Suppose that the radius $R$ of a spherical balloon is increasing at the rate of $4mm/sec$. What is the rate of increase of the balloon’s volume at the point when $R=40\mathrm{mm}$?

05

Related rates - angle of elevation of a balloon

Suppose that a balloon is floating upwards at the rate of $5m/min$. An observer is standing $20\mathrm{m}$ from the release site. After $3\min$, how fast is the viewing angle from observer to balloon increasing?