Sections 11 and 12

This is one of the most awesome review tests that any group of students has ever designed. Not only every single topic in the corresponding portion has been addressed... the questions are evil! Good job, guys. You are getting a bunch of extra credit for this awesome effort.

1. Find the derivative of the following functions

   \begin{align} f(x) &= 230x^{300} + 15x^{18} + 5 \\ f(x) &= -\frac{488}{233} x^{265/488} \\ f(x) &= 2\sin x + 6\cos x + 9 \\ f(x) &= \sin x \cos x \\ f(x) &= 12\tan x + 3x^2 - \cos x \\ f(x) &= \sin^2 x + \cos^2 x \\ f(x) &= e^{8x} \\ f(x) &= 8^x \\ f(x) &= 10^{2x^2} \end{align}

2. Use the chain rule to find the derivative of the following function

   \begin{equation} f(x) = \sqrt{ \big( 3x^5 + \sin(e^x) - 400 \big)^3 } \end{equation}

3. Use the quotient rule to obtain the derivative of

   \begin{equation} f(x) = \frac{x^3}{x^2 + 6} \end{equation}
4. Find the tangent line to the graph of the function \( f(x) = x^3 - x^2 + 60 \) at \( x=2 \).
5. Find the normal line to the graph of function \( f(x) = 4x^2-21x+4 \) at \( x=3 \).

6. Use logarithmic differentiation to obtain the derivative of the following functions

   \begin{align} f(x) &= \big( \cos 8x \big)^x \\ f(x) &= \big( \tan x \big)^{9/x}\\ \end{align}
7. Find all \( x \)-values in the interval \( [0,2\pi] \), where the tangent line to the graph of \( f(x)= \sin^2 x - \cos^2 x \) is horizontal.

8. Use implicit differentiation to obtain the derivative \( y’ \), if \( x \) and \( y \) are related by the following equations:

   \begin{align} e^{x+y} &= y^3 \\ x^2 + y^2 &= 25 \end{align}

Sections 15 and 16

Good assortment of derivatives, although I am missing some hard-core logarithmic differentiation.
The questions on implicit differentiation did not make any sense, and I did not include them.
I wish you would have included some site problems, as well as questions about normal lines.

1. Find the derivative of the following functions:
   \begin{align} f(x) &= 7x + 3x^3-5 \\ f(x) &= \cos x + 8x^2 + \tfrac{1}{2} x^2 +3 \\ f(x) &= x^{5/2} + 3x + 4 \\ f(x) &= \cos^4 x + \cos x^4 \\ f(x) &= 10 e^{\ln x} + 23 e^x \\ f(x) &= \big( 2e^x + x^2 \big)^3 \\ f(x) &= \log (3x+4) \\ f(x) &= 4\log_3 x - \ln x \\ f(x) &= \big( x^{-2} + x^{-3} \big) \big( x^5 -2 x^2 \big) \\ f(x) &= \csc x \cot x \\ f(x) &= 3x^4 + \tan \frac{x}{x-1} \\ f(x) &= \frac{3x^2 e^x}{(x\ln x+1)\ln x}\\ f(x) &= \sin^2 x + \cos^2 x \\ f(x) &= \frac{x\sin x}{\cos x} \\ f(x) &= \sin ( 13x^3-e^x ) \\ f(x) &= \cos(x^2+2) + \tan(3e^x -42\ln x) \end{align}
2. Find the equation of the tangent line to the graph of \( y=f(x) = \dfrac{x+5x^3}{\sqrt{x}} \) at \( x=1 \).