Find the derivative of the following equation in fully simplified form

• Apr 10th 2013, 01:30 PM
digidako
Find the derivative of the following equation in fully simplified form
G(x)=3x * (1-2x^2)^-1/3
I have attached my attempt at the question but I'm sure it's not right.Attachment 27895
• Apr 10th 2013, 01:46 PM
HallsofIvy
Re: Find the derivative of the following equation in fully simplified form
It's pretty close to being right! Your function is $\displaystyle G(x)= 3x(1- 2x^2)^{-1/3}$ and, using the product rule and chain rule, you have
$\displaystyle G'(x)= 3x(-1/3)(1- 2x^2)^{-4/3}(-4x)+ (1- 2x^2)^{-1/3)(3)$, which is correct.

But you appear to be canceling that "-1/3" with the "3" at the end of the formula. You cannot do that because they are not multiplied together. You can cancel the "1/3" with the "3" at the beginning of the fomula: $\displaystyle G'(x)= 4x^2(1- 2x^2)^{-4/3}+ 3(1- 2x^2)^{-1/3}$. You can, in addition, factor $\displaystyle (1- 2x^2)^{-1/3}$ out: $\displaystyle (1- 2x^2)^{-1/3}(4x^2(1- 2x^2)^{-1}+ 3)$.
• Apr 10th 2013, 01:48 PM
Educated
Re: Find the derivative of the following equation in fully simplified form
You have a lot of algebraic mistakes.

In line 2, why did you cross out the 1/3 and 3 (on the right)? They do not cancel out because they are not being multiplied together, notice there is a plus in between them. The 1/3 and the 3 (on the left) can cancel out because they're being multiplied.

In the third line, why did you randomly cross out $\displaystyle (1-2x^2)^{-1/3}$

In the forth line, you cannot bring the $\displaystyle (1-2x^2)^{1/4}$ like that. Can you see your mistake?
• Apr 10th 2013, 02:10 PM
digidako
Re: Find the derivative of the following equation in fully simplified form
Thank you all for your replies. I have used your advice and I have attached my final answer
• Apr 10th 2013, 04:40 PM
Soroban
Re: Find the derivative of the following equation in fully simplified form
Hello, digidako!

Quote:

$\displaystyle \text{Differentiate and simplify: }\:G(x)\:=\:\frac{3x}{(1-2x^2)^{\frac{1}{3}}}$

$\displaystyle G'(x) \;=\;\frac{(1-2x^2)^{\frac{1}{3}}\cdot 3 \:-\:3x\cdot\frac{1}{3}(1-2x^2)^{-\frac{2}{3}}\cdot(-4x)}{(1-2x^2)^{\frac{2}{3}}}$

. . . . .$\displaystyle =\;\frac{3(1-2x^2)^{\frac{1}{3}} + 4x^2(1-2x^2)^{-\frac{2}{3}}}{(1-2x^2)^{\frac{2}{3}}}$

. . . . .$\displaystyle =\;\frac{(1-2x^2)^{-\frac{2}{3}}\cdot\left[3(1-2x^2) + 4x^2\right]}{(1-2x^2)^{\frac{2}{3}}}$

. . . . .$\displaystyle =\;\frac{3-6x^2+4x^2}{(1-2x^2)^{\frac{4}{3}}}$

$\displaystyle G'(x) \;=\;\frac{3-2x^2}{(1-2x^2)^{\frac{4}{3}}}$