## MTALHAHASSAN2 one year ago Determine the second derivative of each of the follwoing:

1. MTALHAHASSAN2

f(x)=sqrt x

2. MTALHAHASSAN2

f1(x)=x^1/2

3. thomas5267

$\sqrt{x}=x^{1/2}$

4. MTALHAHASSAN2

i know

5. MTALHAHASSAN2

but know what about the second derivative

6. MTALHAHASSAN2

@IrishBoy123

7. IrishBoy123

8. MTALHAHASSAN2

f(x)= sqrt x

9. MTALHAHASSAN2

@SolomonZelman

10. MTALHAHASSAN2

wait but how are we goona get the second derivative

11. SolomonZelman

we are going to apply the power rule again.

12. SolomonZelman

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

13. MTALHAHASSAN2

yeah

14. SolomonZelman

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

15. MTALHAHASSAN2

i goted it

16. SolomonZelman

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)

17. SolomonZelman

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