Derivatives and Tangents

• Jan 18th 2009, 04:15 PM
soulmeetsbody
Derivatives and Tangents
Hey, there I have one quick question for now.. I'm in grade 12 calc and we're doing derivatives and have learned the power rule (its a nice shortcut) and i'm having trouble with this one problem:

Write an equation of a tangent to each of the curves at the given point:

$\displaystyle y= \sqrt{3x^3}$ P(3,9)
Heres what I did:

$\displaystyle y=(3x^3)^\frac{1}{2}$
Then the derivative:
...Do I use the power rule on the inside brackets first? or distribute the 1/2 outside the brackets??

$\displaystyle y'=(9x^2)^\frac{1}{2}$
$\displaystyle y'=(3x)$

So my equation turns out to be
9x-y-18=0,
but the answer in the back of the book is 9x-2y-9=0
What am I doing wrong?
• Jan 18th 2009, 04:24 PM
Mush
Quote:

Originally Posted by soulmeetsbody
Hey, there I have one quick question for now.. I'm in grade 12 calc and we're doing derivatives and have learned the power rule (its a nice shortcut) and i'm having trouble with this one problem:

Write an equation of a tangent to each of the curves at the given point:

$\displaystyle y= \sqrt{3x^3}$ P(3,9)
Heres what I did:

$\displaystyle y=(3x^3)^\frac{1}{2}$
Then the derivative:
...Do I use the power rule on the inside brackets first? or distribute the 1/2 outside the brackets??

$\displaystyle y'=(9x^2)^\frac{1}{2}$
$\displaystyle y'=(3x)$

So my equation turns out to be
9x-y-18=0,
but the answer in the back of the book is 9x-2y-9=0
What am I doing wrong?

First of all, you should distribute the power!
$\displaystyle \frac{d}{dx} \sqrt{3x^3} = \frac{d}{dx} (3x^3)^{\frac{1}{2}}$

$\displaystyle = \frac{d}{dx} (3^{\frac{1}{2}}x^{\frac{3}{2}})$

$\displaystyle = \frac{d}{dx} (\sqrt{3}x^{\frac{3}{2}})$

$\displaystyle = \sqrt{3} \frac{d}{dx} x^{\frac{3}{2}}$

$\displaystyle = \sqrt{3} \times \frac{3}{2} \times x^{\frac{1}{2}}$

$\displaystyle = \frac{3\sqrt{3}}{2} x^{\frac{1}{2}}$

$\displaystyle = \frac{3\sqrt{3}}{2}\sqrt{x}$
• Jan 18th 2009, 04:34 PM
soulmeetsbody
Thank you very much! It makes a lot more sense now :)