Ithe problem is as follows: d/dx[1-5x/1+5x] I need to find f'(x) and f''(x)
I found f'(x) but the second derivative is giving me problems. Here is what I did for both: d/dx[1-5x/1+5x] = (1+5x) d/dx [1-5x] - (1-5x) d/dx [1+5x] and all this is divided by (1+5x)^2
= (1+5x)(-5) - (1-5x)(5)/(1+5x)^2 =
-5 -25x - 5 + 25x/(1+5x)^2 =
-10/(1-5x) Now I tried to get f''(x) unsuccessfully I might add.
(1+10x+25x^2) d/dx [-10] - (-10) d/dx [1+10x+25x^2] all this is divided by
(1+10x+25x^2)^2 =
1+10x+25x^2 - (-10)(10)(50x)/(1+10x+25x^2)^2 =
1+5010x+25x^2/(1+10x+25x^2)^2
Please let me know where I went off track.
THANK YOU!
Keith
Hello Keith!
First, I know their was already a post submitted since I've been writing this, but I figured some teachers want you specifically to use the quotient rule over the power rule. Which sucks, because I love the power rule. So using only the quotient rule, your (small) mistakes are below:
To begin, I, without looking at your work, equated f'(x) and got the same answer as you:
f'(x) = -10/[(1+5x)^2]
I think I've found your problem. When you are taking this part of your f''(x):
(1+10x+25x^2) d/dx [-10]
You say that the above argument equals (1+10x+25x^2). But d/dx of -10, or any constant, is 0, not 1. So:
(1+10x+25x^2) d/dx [-10] equals 0 instead of (1+10x+25x^2)
So looking at your work again:
(1+10x+25x^2) d/dx [-10] - (-10) d/dx [1+10x+25x^2] all this is divided by
(1+10x+25x^2)^2 =
1+10x+25x^2 - (-10)(10)(50x)/(1+10x+25x^2)^2 =
1+5010x+25x^2/(1+10x+25x^2)^2
The result would change to:
0 - (-10)(10)(50x)/(1+10x+25x^2)^2 =
5000x/(1+10x+25x^2)^2
The other problem lies at this point in your work:
(-10) d/dx [1+10x+25x^2] = (-10)(10)(50x)
(-10) d/dx [1+10x+25x^2] should equal:
(-10) * (50x+10) which equals: -500x -100
So we now have:
0 - (-500x - 100) / (1+10x+25x^2)^2=
(500x + 100) / (1+10x+25x^2)^2 =
(500x + 100) / (1+5x)^4
Now, if you really want to take it further,
(1 +5x)^4 can be expanded to:
625x^4 + 500x^3 + 150x^2 + 20x + 1
So an optional f''(x) answer is:
(500x+100)/(625x^4 + 500x^3 + 150x^2 + 20x + 1)
But if it were me I'd leave the denominator unexpanded.
So the final answers:
f'(x) = (-10) / (1 + 5x)^2
f''(x) = (500x + 100) / (1 + 5x)^4
Hope this helps!
P.S.
My answer for f''(x) equals the answer from the above post:
To get the above answer, we factor (100) out of the numerator:
(500x +100) / (1 + 5x)^4 = (100)(1 + 5x) / (1 + 5x)^4
The (1 + 5x)'s cancel out, giving us:
100 / (1 + 5x)^3, the answer from the above post.
Hello, Keith!
f(x) .= .(1-5x)(1+5x) . Find f'(x) and f''(x)
Your first derivative is correct: .f'(x) .= .10/(1 + 5x)²
. . but why multiply it out?
EB had the best advice for the second derivative.
You can do it head-on, but you must be more careful.
. . . . . . . (1 + 5x)²·0 - 10·2(1 + 5x)·5 . . . . -100(1 + 5x) . . . . . -100
f''(x) . = . ---------------------------------- . = . --------------- . = . ------------
. . . . . . . . . . . . (1 + 5x)^4 . . . . . . . . . . . .(1 + 5x)^4 . . . . .(1 + 5x)³