# Homework 6

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. Differentiate the following:
1. $f(x) = \sec x$
2. $f(x) = \cot x$
3. $f(x) = \dfrac{\sec x}{1 + \tan x}$
4. $f(t) = e^t\sin t$
5. $f(x) = (x^4 + 4x^2)e^x$
6. $y' = \dfrac{e^x}{1 + x}$
7. $f(t) = \dfrac{t - \sqrt{t}}{t^{1/3}}$
8. $y = e^{\sqrt{x}}$
9. $f(x) = (3x - x^3)^23$
10. $f(x) = \sqrt{1 + \tan x}$
11. $f(x) = xe^{-2x}$
12. $f(x) = (3x + 2)^3(4x^2 + 3)^4$
13. $y = \sqrt{x + \sqrt{ x + \sqrt{x}}}$
14. $f(x) = 2^{\sin \pi x}$
15. $f(x) = \sqrt{\dfrac{x}{x^2 + 4}}$
16. $f(x) = e^{e^x}$
17. $f(x) = \sin x + \sin^2 x$
2. Find the equation of the tangent line to the curve at the given point.
1. $y = (1 + 2x)^{10}, \qquad (0, 1)$
2. $y = \sin(\sin(x)), \qquad (\pi, 0)$
3. Recall the BAC example from 3.1. The concentration function is $C(t) = 0.0225te^{-0.0467t}$ How rapidly is the BAC decreasing a half hour after ingestion?
4. Find $dy/dx$ with implicit differentiation for the equation $x^3 + y^3 + x^2 + y = 1$
5. Consider the equation $x^{2/3} + y^{2/3} = 4$. Find the equation of the tangent line at the point $(-3\sqrt{3}, 1)$.