An archived instance of a Calc II forum

Your most favorite function

mark

(5 pts)

As calculus students, I assume you all have your favorite function. You should tell us what your favorite function is here, using the the forum’s typesetting tools tools for incorporating images. Specifically,

Type in your favorite function so that it’s typeset like this: f(x)=e^{-x^2},
Type in the derivative of your favorite function, so that it’s typeset as well: f'(x) = -2x e^{-x^2}, and finally
Include a graph of both f and f' together.

audrey

My favorite function is f(x) = \sin(x^2) and its derivative is f'(x) = 2x\cos(x^2. The graphs of those things look like:

rstahles

My favorite function is f(x)= \sin(x^4+3) and its derivative is f'(x)= 4x^3\cos(x^4+3). The graph for both of these functions are shown below:

cbrowni1

My favorite function is f(x)=1/{x^2}. The derivative of this function is f'(x)=-2/{x^3}. The graphs of those two functions are shown below.

jlajcin

My favorite function is the function f(x) =cot(x) its derivative is f'(x) = -csc^2 (x)
It looks kinda cool.

agooch

My favorite function is

f(x)=\sin (2x)-2\sin (x)

and its derivative is

f^{\prime }(x)=2\cos (2x)-2\cos (x)

Their graphs look like this!

chowell1

My favorite function is f(x) = sin(x+e^4) and its derivative is f'(x) = cos(x+e^4). Their graphs look like this:

rtaylor4

My favorite function is

f(x)=x^3

and it’s derivative is

f’(x)=3x^2

gbelk

My favorite function is

f(x)=\sin (x^2-3)?

and its derivative is

f^{\prime }(x)=2x\cos (x^2-3)?

the graph for both of these functions are shown below:

audrey

My favorite function is f(x) = x\sin(1/x). The derivative of f is:

f'(x) = \sin\left(\frac{1}{x}\right) - \frac{1}{x^2} x \cos\left(\frac{1}{x}\right).

The graphs of these functions look like so:

afernan2

I am a fan of x^2+y^2=1 because it is the unit circle, very useful.
it’s derivative is 2x+2y=0
It has this graph!

mearing

My favorite function is
f(x)=cos(2x^5)
The derivative is
f'(x)=-10x^4sin(2x^5)
The graph of both functions looks like

myost

function: f(x)=cos(x^2)
derivative: f'(x)=-2x\sin(x^2)
graph:

nhaley

My favorite function is f(x)=\frac{\sin(x^4)}{x}. The derivative of f is:

f(x)=\frac{4x^4\cos(x^4)-\sin(x^4)}{x^2}

The graphs of these functions look like:

knguyen3

My favorite function is the sigmoid function s(x)=1/(1+e^{-x}). Sigmoid function is one of activation functions that is commonly used in neural networks.
It’s derivative is s`(x)=e^{-x}/(1+e^{-x})^2.

fcarrill

My Favorite function is f(x) = \cos(x^2) and its derivative is f'(x) = \cos 2x This is how the graph looks like:

tbrincke

My favorite function is f(x) = cos(x^3)/2
The derivative of this function is f'(x) = -(3x^2sin(x^3))/2

wnovak

My favorite function is the Archimedes Spiral
r=b+a\theta^{1/c}
The derivative of this function is just a circle, but relative to radians it’d be

r=\frac{a\theta^{\frac{\left(1-c\right)}{c}}}{c}

gwebb

My favorite function is:

f(x) = 2\sin\left(x^{2}\right)

and its derivative is:

f'(x) = 4x\cos\left(x^{2}\right)

ccase2

My favorite function is f(x)=ln(sin(x^2)). The derivative of f is:

f(x)=\frac{cos(x^2)2x}{sin(x^2)}

The graph looks like this: