1. ## Differentiation

Find $\displaystyle (\frac{r_1}{|r_2| } )'$for$\displaystyle r_1 = ti + {t}^{ 2} j - 3tk$and $\displaystyle r_2 = {t}^{ 2}i - 2j + tk$

I have found $\displaystyle |r_2| = \sqrt{{t}^{ 4} + {t}^{2 } + 4 }$

But now when i differntaite i do not get there answer of

$\displaystyle \frac{4-{t}^{4}i}{({t}^{ 4} + {t}^{2 } + 4)^\frac{3}{2 }} }+ \frac{{t}^{3}+ 8tj}{({t}^{ 4} + {t}^{2 } + 4)^\frac{3}{2 }} + \frac{3({t}^{4}-4k}{({t}^{ 4} + {t}^{2 } + 4)^\frac{3}{2 } }$

Find $\displaystyle (\frac{r_1}{|r_2| } )'$for$\displaystyle r_1 = ti + {t}^{ 2} j - 3tk$and $\displaystyle r_2 = {t}^{ 2}i - 2j + tk$

I have found $\displaystyle |r_2| = \sqrt{{t}^{ 4} + {t}^{2 } + 4 }$

But now when i differntaite i do not get there answer of

$\displaystyle \frac{4-{t}^{4}i}{({t}^{ 4} + {t}^{2 } + 4)^\frac{3}{2 }} }+ \frac{{t}^{3}+ 8tj}{({t}^{ 4} + {t}^{2 } + 4)^\frac{3}{2 }} + \frac{3({t}^{4}-4k}{({t}^{ 4} + {t}^{2 } + 4)^\frac{3}{2 } }$
Well we can certainly verify that their answer is correct. What is your work? What did you get and how did you get it? We cannot help you unless you tell us what you did.

-Dan

3. Originally Posted by topsquark
Well we can certainly verify that their answer is correct. What is your work? What did you get and how did you get it? We cannot help you unless you tell us what you did.

-Dan
Ok so $\displaystyle \frac{ti + {t}^{ 2} j - 3tk}{({t}^{4 }+{t}^{2 } +4)^\frac{1}{ 2} }$ so first i will just differentiate i $\displaystyle \frac{t}{({t}^{4 }+{t}^{2 } +4)^\frac{1}{ 2} } } = t({t}^{4 }+{t}^{2 } +4)^\frac{-1}{ 2} = \frac{-\frac{1}{2 } t({4t}^{3 } +2t)}{({t}^{4 }+{t}^{2 } +4)^\frac{3}{ 2} }$

Ok so $\displaystyle \frac{ti + {t}^{ 2} j - 3tk}{({t}^{4 }+{t}^{2 } +4)^\frac{3}{ 2} }$ so first i will just differentiate i $\displaystyle \frac{t}{({t}^{4 }+{t}^{2 } +4)^\frac{3}{ 2} } } = t({t}^{4 }+{t}^{2 } +4)^\frac{-1}{ 2} = \frac{-\frac{1}{2 } t({4t}^{3 } +2t)}{({t}^{4 }+{t}^{2 } +4)^\frac{3}{ 2} }$
First, the exponent on the t^4 + t^2 + 4 before you take the derivative is 1/2 not 3/2.

Your notation is excrescent. Don't simply put equal signs between different steps of a problem.
$\displaystyle \frac{t}{({t}^{4 }+{t}^{2 } +4)^\frac{3}{ 2} } } = t({t}^{4 }+{t}^{2 } +4)^\frac{-1}{ 2} = \frac{-\frac{1}{2 } t({4t}^{3 } +2t)}{({t}^{4 }+{t}^{2 } +4)^\frac{3}{ 2} }$

None of these are equal.

You wish to find the derivative of
$\displaystyle \frac{t}{(t^4 + t^2 + 4)^{1/2}}$

Use the quotient rule:
$\displaystyle \frac{d}{dt} \left ( \frac{t}{(t^4 + t^2 + 4)^{1/2}} \right ) = \frac{(1)(t^4 + t^2 + 4)^{1/2} - t(1/2)(t^4 + t^2 + 4)^{-1/2}(4t^3 + 2t)}{t^4 + t^2 + 4}$

Now simplify.

-Dan