1. ## Derivative question

If f(x) = sqrt(3x)
then f'(x) = ?
and f' (4) = ?
I have done this problem over and over again and the result remains the same, incorrect. Here is what I did:
d/dx[sqrt(3x)] =
d/dx[sqrt(3)] d/dx[x^1/2] =
0 + 1/2x^-1/2

I also tried 3^1/2 + 1/2x^-1/2 = 1/2 +1/2/sqrt(x)
I must not using the rules correctly and I read them over again and I am not getting it. Please explain the error of my ways.
Thankx,
Keith

2. Originally Posted by kcsteven
If f(x) = sqrt(3x)
then f'(x) = ?
and f' (4) = ?
I have done this problem over and over again and the result remains the same, incorrect. Here is what I did:
d/dx[sqrt(3x)] =
d/dx[sqrt(3)] d/dx[x^1/2] =
0 + 1/2x^-1/2

I also tried 3^1/2 + 1/2x^-1/2 = 1/2 +1/2/sqrt(x)
I must not using the rules correctly and I read them over again and I am not getting it. Please explain the error of my ways.
Thankx,
Keith

Okay, here we go!

sqrt(3x) is the same as (3x)^1/2
So use the power rule as you would any parenthetical polynomial under an exponent.

Just as the derivative of (2x + 7)^3 is (3) (2x + 7)^2 (2),

the derivative of (3x)^1/2 is (1/2) (3x)^-1/2 (3).

The (1/2) is the old parenthetical exponent, the (3x)^-1/2 is acquired by doing the power rule and subtracting one from the exponent, and the (3) comes from taking the derivative of the parenthetical argument, 3x.

So we end up with (3/2) (3x)^-1/2
Or (3/2) (sqrt(3x))^-1
Or (3/2)/(sqrt(3x))
Which, in taking the sqrt out of the denominator for simplest radical form by multiplying the numerator and denominator by sqrt(3x), (if you prefer), is:

[(3/2)(sqrt(3x))] / 3x
As a final move, the 3/2 and 3 reduce to 1 and 2 respectively, making the final answer:

sqrt(3x) / 2x

Hope this helps!

3. Originally Posted by kcsteven
If f(x) = sqrt(3x)
then f'(x) = ?
and f' (4) = ?
I have done this problem over and over again and the result remains the same, incorrect. Here is what I did:
d/dx[sqrt(3x)] =
d/dx[sqrt(3)] d/dx[x^1/2] =
0 + 1/2x^-1/2

I also tried 3^1/2 + 1/2x^-1/2 = 1/2 +1/2/sqrt(x)
I must not using the rules correctly and I read them over again and I am not getting it. Please explain the error of my ways.
Thankx,
Keith

Sorry! I didn't answer the last question!
So as I said above,
f'(x) = sqrt(3x) / 2x
So f'(4) = sqrt(12) / 8

Which equals sqrt(4)sqrt(3) / 8
Which equals 2sqrt(3) / 8
Which simplifies to sqrt(3) / 4

So:

f'(x) = sqrt(3x) / 2x
f'(4) = sqrt(3) / 4

If anyone can verify this, it would be lovely, but I'm pretty confident these are correct. Again, hope this helps!

4. Originally Posted by Pajkaj
If anyone can verify this, it would be lovely, but I'm pretty confident these are correct. Again, hope this helps!
You got it right on all counts!

-Dan