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Algebra Help for a Derivative

There's probably easier ways to solve this derivative by using logarithmic rules, but using the method that I took, do you see where I go wrong with the algebra?

It would greatly help to see what errors I making! I think the same errors pop up when doing problems like this. Thank you!

Re: Algebra Help for a Derivative

I can't really read what you wrote. It looks like the problem is $\displaystyle y = \ln\left( \dfrac{(x^2+1)^5}{\sqrt{2-x}} \right)$. Then, when you get to line 6, it appears that $\displaystyle \dfrac{1}{\left(\dfrac{(x^2+1)^5}{\sqrt{2-x}}\right) }$ becomes $\displaystyle \dfrac{1}{\sqrt{2-x} (x^2+1)^5}\right) }$. That is wrong. It should become $\displaystyle \dfrac{\sqrt{2-x}}{(x^2+1)^5}$

Re: Algebra Help for a Derivative

Yes! That's what puzzles me a lot. I don't have that part down in my head yet.

When dividing a complex fraction like that, I confuse the order of *what group* divides what.

Any advice on that? I took it that 1 divides (x^2 +1)^5 first.

Re: Algebra Help for a Derivative

Multiply numerator and denominator by $\displaystyle \sqrt{2-x}$. You get $\displaystyle 1\cdot \sqrt{2-x}$ in the numerator and $\displaystyle \dfrac{(x^2+1)^5}{\sqrt{2-x}} \sqrt{2-x}$ in the denominator. The $\displaystyle \sqrt{2-x}$ and the $\displaystyle \dfrac{1}{\sqrt{2-x}}$ cancel each other out.

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Re: Algebra Help for a Derivative

Re: Algebra Help for a Derivative

Looks good with a quick glance

Re: Algebra Help for a Derivative

Quote:

Originally Posted by

**IdentityProblem** There's probably easier ways to solve this derivative by using logarithmic rules, but using the method that I took, do you see where I go wrong with the algebra?

It would greatly help to see what errors I making! I think the same errors pop up when doing problems like this. Thank you!

Why don't you use the logarithm laws to make life easier on yourself?

$\displaystyle \displaystyle \begin{align*} \ln{ \left[ \frac{ \left( x^2 + 1 \right) ^5 }{\sqrt{1 - x} } \right] } &= \ln{ \left[ \left( x^2 + 1 \right) ^5 \right] } - \ln{ \left( \sqrt{1 - x} \right) } \\ &= 5\ln{ \left( x^2 + 1 \right) } - \ln{ \left[ \left( 1 - x \right) ^{\frac{1}{2}} \right] } \\ &= 5 \ln{ \left( x^2 + 1 \right) } - \frac{1}{2} \ln{ \left( 1 - x \right) } \end{align*}$

These functions are MUCH easier to differentiate...