Differential dy problem

[torque]

How do I find the differential dy to the function y = x(sqrt(1-x^2)) ?


I got it down to y = x(1-x^2) ^ 1/2 but I'm stuck from there. Am I supposed to use the chain rule here for the product rule?

 

[Deleted member 4993]

How do I find the differential dy to the function y = x(sqrt(1-x^2)) ?


I got it down to y = x(1-x^2) ^ 1/2 but I'm stuck from there. Am I supposed to use the chain rule here for the product rule?

$\displaystyle y \ = \ x * (1 - x^2)^{\frac{1}{2}}$

$\displaystyle \frac{dy}{dx} \ = \ 1 * (1 - x^2)^{\frac{1}{2}} + x * \frac{1}{2} * (1 - x^2)^{-\frac{1}{2}} * (-2 * x)$

Now simplify the above......

 

[torque]

I got sqrt(1 - x^2) - x^2 / sqrt(1 - x^2)


My book's correct answer is: 1 - 2x^2 / sqrt(1 - x^2)

What did I do wrong?

 

[galactus]

Use proper grouping symbols. They mean everything. You have $\displaystyle 1-\frac{2x^{2}}{\sqrt{1-x^{2}}}$ written, when I think you mean

$\displaystyle \frac{1-2x^{2}}{\sqrt{1-x^{2}}}$

You're not wrong. It's the same thing. The book found a common denominator and put it in a different form

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