Indicated derivative and limits

[Cees Schouten]

Hi,

I need some help figuring out the following questions, steps would be much appreciated.

1) Find the indicated derivative.
dy/dx if y = 9 / root (x + 8 )
Type an exact answer, using radicals as needed (or so does it say)

2) Determine lim h→ 0 f (x+h) - f (x) / h if f(x) = root (20x+37)

Thanks in advance!

 

[HallsofIvy]

Hi,

I need some help figuring out the following questions, steps would be much appreciated.

1) Find the indicated derivative.
dy/dx if y = 9 / root (x + 8 )
Type an exact answer, using radicals as needed (or so does it say)

2) Determine lim h→ 0 f (x+h) - f (x) / h if f(x) = root (20x+37)

Thanks in advance!

It's really just some basic algebra to start with.

1) \(\displaystyle \frac{9}{\sqrt{x+8}}= 9(x+ 8)^{-1/2}\). Now, do you know how to find the derivative of \(\displaystyle x^n\)?

2) \(\displaystyle f(x+h)= \sqrt{20x+ 37}\) so \(\displaystyle f(x+h)= \sqrt{20(x+h)+ 37}= \sqrt{20x+ 20h+ 37}\)

 

[Cees Schouten]

1) [FONT=MathJax_Main]9[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]8[/FONT][FONT=MathJax_Main]√[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]9[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]8[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Main]2[/FONT]. Now, do you know how to find the derivative of [FONT=MathJax_Math]x[/FONT][FONT=MathJax_Math]n[/FONT]?

Is it -9/(2 (8+x)^(3/2))?

2) I still wouldnt have a clue what I'm supposed to do. Is it like this?


[FONT=MathJax_Main]




[/FONT]

 

[HallsofIvy]

1) [FONT=MathJax_Main]9[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]8[/FONT][FONT=MathJax_Main]√[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]9[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]8[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Main]2[/FONT]. Now, do you know how to find the derivative of [FONT=MathJax_Math]x[/FONT][FONT=MathJax_Math]n[/FONT]?

Is it -9/(2 (8+x)^(3/2))?

Awkwardly written but correct. Is there any reason for swapping x+ 8 to 8+ x? It's the same thing, of course, but I wonder why you did it.

2) I still wouldnt have a clue what I'm supposed to do. Is it like this?

View attachment 2223

You were asked to find (f(x+h)- f(x))/h. Where did that first product come from?
[FONT=MathJax_Main]

[/FONT]

 

[Cees Schouten]

Awkwardly written but correct. Is there any reason for swapping x+ 8 to 8+ x? It's the same thing, of course, but I wonder why you did it.

I think it's to blame to me not working precisely. You said it's awkwardly written but I can't seem to find a more elegant solution. Is there one?

You were asked to find (f(x+h)- f(x))/h. Where did that first product come from?

[FONT=MathJax_Main]
I used your version of f(x+h) to fill in the formula.

[/FONT]lim h→ 0 f (x+h) - f (x) / h if f(x) = root (20x+37).
Is this exactly the definition of df/dx? So this limit is equal?
http://www.wolframalpha.com/share/clip?f=d41d8cd98f00b204e9800998ecf8427evecqte0e4o

 

[HallsofIvy]

I think it's to blame to me not working precisely. You said it's awkwardly written but I can't seem to find a more elegant solution. Is there one?



[FONT=MathJax_Main]
I used your version of f(x+h) to fill in the formula.

Oh, dear! Blush, bush. There was supposed to be an "=" between those! I have gone back and corrected my post.

[/FONT]

lim h→ 0 f (x+h) - f (x) / h if f(x) = root (20x+37).
Is this exactly the definition of df/dx? So this limit is equal?

Yes that is the derivative of "derivative".

 

[Cees Schouten]

Oh, dear! Blush, bush. There was supposed to be an "=" between those! I have gone back and corrected my post.

[/FONT][/COLOR][/COLOR]

Yes that is the derivative of "derivative".

2) So the limit is 10/root(37+20x)?

1) Sorry to repeat myself, but is there a more elegant solution than -9/(2 (8+x)^(3/2))?

 

[JeffM]

2) So the limit is 10/root(37+20x)?

1) Sorry to repeat myself, but is there a more elegant solution than -9/(2 (8+x)^(3/2))? Not really. Halls of Ivy was just asking why you reversed

(x + 8) in the original problem to (8 + x). Perhaps, it is SLIGHTLY more elegant to write: \(\displaystyle - \dfrac{9}{2\sqrt{(x + 8)^3}}.\)

It still looks pretty ugly to me, but what is important is that it is correct.

 

[JeffM]

2) So the limit is 10/root(37+20x)?

That is the correct derivative, but you still have not shown how you reached it, which was the original problem.