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For ${\displaystyle f\left(t\right)=(\frac{t}{e^{1-3t}},2t^2-t)}$ what is the distance between f(2) and f(5)?

Differentiate the parametric function and find dy/dx and $d^{2}y/d{x}^2$
Differentiate the parametric function and find $\frac{\mathrm{d}y}{\mathrm{d}x}$ and $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}$ in terms of "t" when:
$x=\frac{1}{t-<!--\; -\; -->1}$ and $y=\frac{1}{t+1}$
I have first started by finding $\frac{\mathrm{d}y}{\mathrm{d}x}$ by finding $\frac{\mathrm{d}x}{\mathrm{d}t}$ and $\frac{\mathrm{d}y}{\mathrm{d}t}$ which comes to
$\frac{\mathrm{d}x}{\mathrm{d}t}=\ln <!--\; \; -->|t-<!--\; -\; -->1|$
and $\frac{\mathrm{d}y}{\mathrm{d}t}=\ln <!--\; \; -->|t+1|$
and used $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{d}y/\mathrm{d}t}{\mathrm{d}x/\mathrm{d}t}$, which gives
$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\ln <!--\; \; -->|t+1|}{\ln <!--\; \; -->|t-<!--\; -\; -->1|}$ now if I divide them by each other doesn't it equal to 0? What have I done wrong here?

How do you write the cartesian equation for $x=t-<!--\; -\; -->2$ and $y=-<!--\; -\; -->(t^2)+t+2?$

How do you find the angle between the planes $2x+5y-<!--\; -\; -->z=6$ and $3x-<!--\; -\; -->2y+6z=10$ ?

What is the polar form of ${\displaystyle (-1,121)}$ ?

What is the polar form of ${\displaystyle (4,32)}$ ?

For ${\displaystyle f\left(t\right)=(\ln t-e^t,\frac{t^2}{e^t})}$ what is the distance between f(2) and f(4)?

What is the distance between the following polar coordinates?:
${\displaystyle (6,\frac{17\pi }{12}),(5,\frac{9\pi }{8})}$

What is the polar form of ${\displaystyle (4,-2)}$ ?

Converting between explicit function and parametric function
Given an explicit function $y=f(x)$, how to convert it to the respective parametric functions $x=f_1(t)y=f_2(t)$ ?
Given parametric functions $x=f_1(t)y=f_2(t)$, how to obtain the respective implicit function $f(x,y)=0$ ?

How do you find the equation of the tangent line to the curve given by parametric equations: ${\displaystyle x=1+\left(\frac{1}{t^2}\right)}$, ${\displaystyle y=1-\left(\frac{3}{t}\right)}$ at the point when t=2?

Parametric Function for Coloring
I'm trying to write a parametric straightline function that changes its values between 0.529 and 0.933.
Usually what I would do is:
$r=0.529+(0.933-<!--\; -\; -->0.529)\ast <!--\; \ast \; -->v$ where parameter $v=[0,1]$
This ensure that my r value will either be 0.529 or 0.933
However, due to some limitations my parameter v has to be changed to $v=[-<!--\; -\; -->2,1]$
Is there anyway that I can change the straightline function such that it will still return r as either 0.529 or 0.933?

What is the polar form of (19, 55)?

Getting a differential equation from a parametric function family
$$\begin{gathered} x=c{e}^{-<!--\; -\; -->t}-<!--\; -\; -->2t+2 \\ y=c(1+t)e^{-<!--\; -\; -->t}-<!--\; -\; -->t^2+2 \end{gathered}$$

For ${\displaystyle f\left(t\right)=(\sin 2t,{\cos }^{2}t)}$ what is the distance between ${\displaystyle f\left(\frac{\pi }{4}\right)}$ and ${\displaystyle f\left(\pi \right)}$

What is the polar form of ${\displaystyle (11,-8)}$ ?

What is the distance between the following polar coordinates?:
${\displaystyle (10,\frac{17\pi }{12}),(4,\frac{15\pi }{8})}$

What is the distance between the following polar coordinates?:
${\displaystyle (-3,\frac{7\pi }{4}),(5,\frac{5\pi }{8})}$

How do you write the corresponding rectangular equation by eliminating the parameter given $x=3t-<!--\; -\; -->1,y=2t+1$?

How do you write a vector equation and a parametric equation for each line: the line through $A(1,-<!--\; -\; -->3,1)$ and parallel to vector $u=(2,-<!--\; -\; -->2,1)$ ?