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?