Question

Determine the second derivative of each of the follwoing:

Answer 1

f(x)=sqrt x

Answer 2

f1(x)=x^1/2

Answer 3

\[
\sqrt{x}=x^{1/2}
\]

Answer 4

i know

Answer 5

but know what about the second derivative

Answer 6

@IrishBoy123

Answer 7

what is your starting equation?

Answer 8

f(x)= sqrt x

Answer 9

@SolomonZelman

Answer 10

wait but how are we goona get the second derivative

Answer 11

we are going to apply the power rule again.

Answer 12

We apply the power rule twice, that is all.
(Want an example?)

Answer 13

yeah

Answer 14

Ok, just tell me what do you get for the derivative of \(x^{1/2}\), when you apply the power rule?

Answer 15

i goted it

Answer 16

Yeah, IrishBoy, lol. I indeed made the biggest mistake in the world. 9I guess the integration power rule got me mixed up just a bit)

Answer 17

The first derivative of \(\large\color{black}{ \displaystyle x^n }\) is given by the power rule:
\(\large\color{black}{ \displaystyle \frac{d}{dx}\left[x^n\right]=nx^{n-1} }\)

and then the second derivative of that would be:
\(\large\color{black}{ \displaystyle \frac{d^2}{dx^2}\left[x^n\right]=\frac{d}{dx}\left[nx^{n-1}\right]=n(n-1)x^{n-2} }\)

perhaps there are a few exceptions to this rule:
\(n\ne 1\) and \(n\ne 0\)


*++++++++++++++++++++++++++++++*

\(\Large\color{black}{ \displaystyle f(x)=x^{\frac{1}{5}} }\)

the first derivative, using the power rule is as follows:

\(\Large\color{black}{ \displaystyle f'(x)=\left(\frac{1}{5}\right)x^{\frac{1}{5}-1} =\frac{1}{5}x^{-\frac{4}{5}}}\)

Then, the second derivative you would find by differentiating f'(x)
again, using the power rule.

\(\Large\color{black}{ \displaystyle f''(x)=\left(\frac{1}{5}\right)\left(-\frac{4}{5}\right)x^{-\frac{4}{5}-1}}\)

and this simplifies to:

\(\Large\color{black}{ \displaystyle f''(x)=-\frac{4}{5}x^{-\frac{9}{5}}}\)