**Directions:**

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

Answer the following:

- Find $dy/dx$ with implicit differentiation:
- $x^2 - xy - y^2 = 4$
- $1 + x = \sin(xy)$
- $2\sqrt{x} + \sqrt{y} = 3$
- $(x + y)^3 + x^3 + y^3 = 0$

- Find the equation of the tangent line at $(1,2)$ for the equation \[x^2 + 2xy - y^2 + x = 2\]
- 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?
- 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.
- Differentiate the following. Remember you can use logarithmic differentiation to simplify the form for some problems.
- $f(x) = x\ln x$
- $f(x) = \ln (\sin^2 x)$
- $f(x) = \sqrt[5]{\ln x}$
- $f(x) = \ln{\sqrt[5]{x}}$
- $y = \ln(e^{-x} + xe^{-x})$
- $y = \dfrac{\sin x \cos^2 x}{(x^2 + 1)^4}$
- $f(x) = x\ln x$
- $f(x) = x^{\cos x}$
- $f(x) = (\sin x)^{\sin x}$

- Find the linearization of $f(x) = \cos x$ at $a = \pi /2$.
- Use a linear approximation to estimate $e^{-0.0015}$.