the slop is not defined

[al-horia]

hi
In my question
I have to find the equation
my problem is in slop
when I try to find the slop I get it not defined

is not defined

 

[soroban]

Hello, al-horia!

The word is slope.

I have to find slope at \(\displaystyle x = \text{-}1\)

. . . \(\displaystyle f(x) \:=\:\sqrt[3]{\dfrac{x}{x+2}} \)

We have: .\(\displaystyle f(x) \;=\;\left(\dfrac{x}{x+2}\right)^{\frac{1}{3}} \;=\;\dfrac{x^{\frac{1}{3}}}{(x+2)^{\frac{1}{3}}} \)

Quotient Rule: .\(\displaystyle f'(x) \;=\;\dfrac{(x+2)^{\frac{1}{3}}\!\cdot\!\frac{1}{3}x^{\text{-}\frac{2}{3}} - x^{\frac{1}{3}}\!\cdot\!\frac{1}{3}(x+2)^{\text{-}\frac{2}{3}}}{(x+2)^{\frac{2}{3}}} \)

Factor: .\(\displaystyle f'(x) \;=\;\dfrac{\frac{1}{3}x^{\text{-}\frac{2}{3}}(x+2)^{\text{-}\frac{2}{3}}\,\big[(x+2) - x\big]}{(x+2)^{\frac{2}{3}}} \)
. . . . . .\(\displaystyle f'(x) \;=\;\dfrac{2}{3x^{\frac{2}{3}}(x+2)^{\frac{4}{3}}}\)


Therefore: .\(\displaystyle f'(\text{-}1) \;=\;\dfrac{2}{3(\text{-}1)^{\frac{2}{3}}(\text{-}1+2)^{\frac{4}{3}}} \;=\; \dfrac{2}{3\!\cdot\!1\!\cdot\!1} \;=\;\dfrac{2}{3}\)

 

[al-horia]

Hello, al-horia!

The word is slope.


We have: .\(\displaystyle f(x) \;=\;\left(\dfrac{x}{x+2}\right)^{\frac{1}{3}} \;=\;\dfrac{x^{\frac{1}{3}}}{(x+2)^{\frac{1}{3}}} \)

Quotient Rule: .\(\displaystyle f'(x) \;=\;\dfrac{(x+2)^{\frac{1}{3}}\!\cdot\!\frac{1}{3}x^{\text{-}\frac{2}{3}} - x^{\frac{1}{3}}\!\cdot\!\frac{1}{3}(x+2)^{\text{-}\frac{2}{3}}}{(x+2)^{\frac{2}{3}}} \)

Factor: .\(\displaystyle f'(x) \;=\;\dfrac{\frac{1}{3}x^{\text{-}\frac{2}{3}}(x+2)^{\text{-}\frac{2}{3}}\,\big[(x+2) - x\big]}{(x+2)^{\frac{2}{3}}} \)
. . . . . .\(\displaystyle f'(x) \;=\;\dfrac{2}{3x^{\frac{2}{3}}(x+2)^{\frac{4}{3}}}\)


Therefore: .\(\displaystyle f'(\text{-}1) \;=\;\dfrac{2}{3(\text{-}1)^{\frac{2}{3}}(\text{-}1+2)^{\frac{4}{3}}} \;=\; \dfrac{2}{3\!\cdot\!1\!\cdot\!1} \;=\;\dfrac{2}{3}\)

thanks dear

 

[Deleted member 4993]

hi
In my question
I have to find the equation
my problem is in slop
when I try to find the slop I get it not defined

is not defined

Is your function:

\(\displaystyle f(x) = \ \sqrt{\left (\dfrac{x}{x+2}\right )^3}\) ...........................................(1)

or

\(\displaystyle f(x) = \ ^3\sqrt{\left (\dfrac{x}{x+2}\right )}\)............................................(2)

If it is the second - then your f'(x) is incorrect.