# Thread: Find the derivative of the following equation in fully simplified form

1. ## 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.

2. ## Re: Find the derivative of the following equation in fully simplified form

It's pretty close to being right! Your function is $G(x)= 3x(1- 2x^2)^{-1/3}$ and, using the product rule and chain rule, you have
$G'(x)= 3x(-1/3)(1- 2x^2)^{-4/3}(-4x)+ (1- 2x^2)^{-1/3)(3)$, 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: $G'(x)= 4x^2(1- 2x^2)^{-4/3}+ 3(1- 2x^2)^{-1/3}$. You can, in addition, factor $(1- 2x^2)^{-1/3}$ out: $(1- 2x^2)^{-1/3}(4x^2(1- 2x^2)^{-1}+ 3)$.

3. ## 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 $(1-2x^2)^{-1/3}$

In the forth line, you cannot bring the $(1-2x^2)^{1/4}$ like that. Can you see your mistake?

5. ## Re: Find the derivative of the following equation in fully simplified form

Hello, digidako!

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

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

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

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

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

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