# Homework 7

Directions:

1. Show each step of your work and fully simplify each expression.
2. Turn in your answers in class on a physical piece of paper.
3. Staple multiple sheets together.
4. Feel free to use Desmos for graphing.

1. Find $dy/dx$ with implicit differentiation:
1. $x^2 - xy - y^2 = 4$
2. $1 + x = \sin(xy)$
3. $2\sqrt{x} + \sqrt{y} = 3$
4. $(x + y)^3 + x^3 + y^3 = 0$
2. Find the equation of the tangent line at $(1,2)$ for the equation $x^2 + 2xy - y^2 + x = 2$
3. The formula for the volume $V$ of a cube with side length $s$ is $V = s^3$. The sides of the cube are 5 feet long and increasing at the rate of $0.2$ inches/second. How fast is the volume of the cube changing?
4. Suppose a drop of dye was dropped into a large bowl filled with water. If the dye spreads out in a circle and its radius is increasing at a rate of 2 cm/sec, determine how fast the area is increasing when the radius of the circle is 10 cm.
5. Differentiate the following. Remember you can use logarithmic differentiation to simplify the form for some problems.
1. $f(x) = x\ln x$
2. $f(x) = \ln (\sin^2 x)$
3. $f(x) = \sqrt{\ln x}$
4. $f(x) = \ln{\sqrt{x}}$
5. $y = \ln(e^{-x} + xe^{-x})$
6. $y = \dfrac{\sin x \cos^2 x}{(x^2 + 1)^4}$
7. $f(x) = x\ln x$
8. $f(x) = x^{\cos x}$
9. $f(x) = (\sin x)^{\sin x}$
6. Find the linearization of $f(x) = \cos x$ at $a = \pi /2$.
7. Use a linear approximation to estimate $e^{-0.0015}$.