## Practice for second midterm

This is selection of problems on derivatives (basic stuff, no applications) for the training sessions before the second midterm. Some problems might be similar to the ones in the exam, some others completely equal, and some others completely different. Remember: the problems in the actual exam come from the HW assigned on the lesson plan of the syllabus. The objective of this page is to foster cooperation to solve as many as possible among yourselves. Use the forum to ask for directions, and even offer help to other students that may ask similar questions. I will be happy to participate in the forum, giving hints in the right direction, as my schedule permits.

1. Use the definition of limit to compute the derivative of the following functions:
$f(x)=\sqrt{x}, g(x)=x^{-1/2}, h(x)=e^x.$
2. Compute the tangent line to the graph of $y=3e^x+\sin x$ at $x=0.$
3. At what values of $x$ is the tangent line of the graph of $y=e^x \cos x$ horizontal?
4. Use logarithmic differentiation to compute the derivative of the following function:
$f(x)= (3x^2+4x-5)^7 (6x^{-2} - 3\sqrt{x})^5.$
5. At what values of $x$ is the tangent line to the graph of $y=4x^3-3x$ parallel to the line $y=3+8x$?
6. Compute $\frac{dy}{dx}$ if $\sin(x+y)=y^2 \cos x.$
7. Find the derivative of the following functions:
 $f(x)=5x$ $f(x)=5(x^2+3x-4)^{10}$ $f(x)=\frac{5}{x}$ $f(x)=\displaystyle{\frac{5}{x^2+x+1}}$ $f(x)=12e^x$ $f(x)=12e^{3x^2+7x}$ $f(x)=12e^{\sin x}$ $f(x)=\displaystyle{\frac{12}{x e^{\sin x}}}$ $f(x)=9\ln x$ $f(x)=9\ln (\tan x)$ $f(x)=9\ln (\ln \tan x)$ $f(x)= \displaystyle{\frac{9}{\ln( \ln \tan x)}}$ $f(x) = 14\sin x$ $f(x) = 15 \cos (3x^2-\sqrt{x})$ $f(x) = 16 \tan\big( \cos (x^2+3) \big)$ $f(x) = \displaystyle{\frac{17 x \sin x}{\tan \big( \cos (x^2+3) \big)}}$