Find the derivative of the function

[frosty8688]

1. Find the derivative of the function using the quotient rule and also by simplifying



2. F(x) = (x - 3x√x)/√x



3. (9x^2 - x - 8) / (2x √x)

 

[eumyang]

You need to show us your attempt at working these out before we can help you.

 

[frosty8688]

$\sqrt{x}$ (1-3(1$/$2$\sqrt{x}$)) - (x-3x$\sqrt{x}$)(1$/$2$\sqrt{x}$)$/$$\sqrt{x}$$^{2}$ = (1-9-x+9x$^{2}$)$/$(2$\sqrt{x}$$\sqrt{x}$$^{2}$) = (9x$^{2}$-x-8) $/$ (2x$\sqrt{x}$)

 

[Mentallic]

$$\frac{d}{dx}x^n=nx^{n-1}$$
And so for the numerator you should have,
$$\frac{d}{dx}\left(x-3x\sqrt{x}\right)$$
$$=\frac{d}{dx}\left(x-3x^{3/2}\right)$$
$$=1-3\left(\frac{3}{2}\cdot x^{1/2}\right)$$

 

[HallsofIvy]

Personally, I wouldn't have used the quotient rule for this at all. I would have written the function as $F(x)= (x+ x^{3/2})x^{-1/2}= x^{1/2}+ x$

The other function is $(9x^2- x- 8)(1/2)(x^{-3/2}= (1/2)(9x^{1/2}- x^{-1/2}- 9x^{-3/2})$

 

[frosty8688]

It said to use both simplification and the quotient rule and to show that the two are the same.

 

[frosty8688]

Personally, I wouldn't have used the quotient rule for this at all. I would have written the function as $F(x)= (x+ x^{3/2})x^{-1/2}= x^{1/2}+ x$

The other function is $(9x^2- x- 8)(1/2)(x^{-3/2}= (1/2)(9x^{1/2}- x^{-1/2}- 9x^{-3/2})$

It said to use both simplification and the quotient rule and to show that the answers are the same.

 

[frosty8688]

Here's what I got for the quotient rule: $\frac{1}{2}$(x$^{-1}$ - x$^{-1/2}$)

 

[frosty8688]

For simplification, I get: (x-3x$^{3/2}$) x$^{-1/2}$ = (x-$\frac{9}{2}$x$^{1/2}$) x$^{-1/2}$

 

[micromass]

Please show us the actual steps you went through. What did you do to obtain those results?