Thread: First and Second Derivatives

1. First and Second Derivatives

Find the first and second derivatives of the function y=4x/ square root (X+1)

Thanks!

2. Hello,
Originally Posted by midsummer
Find the first and second derivatives of the function y=4x/ square root (X+1)

Thanks!
$\displaystyle f(x)=\frac{4x}{\sqrt{x+1}}=4x \cdot (x+1)^{-1/2}$ (remember that sqrt(x)=x^(1/2))

Note that the constant is not important, because $\displaystyle [af(x)]'=af'(x)$ for any constant.

Apply product rule :
$\displaystyle f'(x)=4 (x+1)^{-1/2}+4x \cdot \tfrac{-1}{2} \cdot (x+1)^{-1/2-1}$$\displaystyle =4(x+1)^{-1/2}-2x (x+1)^{-3/2}=2(x+1)^{-3/2} \big[2(x+1)-x\big]$

$\displaystyle f'(x)=2(x+1)^{-3/2} \big[x+2\big]$

Apply product rule again to get $\displaystyle f''(x)$

3. You will have to use the quotient rule for this question.

The quotient rule is $\displaystyle \frac{d}{dx} \frac{f(x)}{g(x)} = \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2}$.

We have: $\displaystyle y = \frac{4x}{\sqrt{x+1}}$

Therefore, $\displaystyle y' = \frac{\sqrt{x+1} \cdot 4 - 4x \cdot (1/2)(x+1)^{-1/2}}{x + 1} = \frac{4(x + 1) - 2x}{(x + 1)^{3/2}} = \frac{2x + 4}{(x + 1)^{3/2}}$.

This is the first derivative.

To obtain the second derivative, you must use the quotient rule to determine the derivative of the first derivative.

4. thanks...and could you demonstrate how to find the second derivative? I understand the concept and I am trying to simplify (it's been a while since I have done this stuff and need lots of practice). again I really appreciate the help!

Mariam