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If y = x3  the x = y1/3

Find the derivative of dy/dx and dx/dy, and henc show that dy/dx * dx/dy = 1.

 

My derivatives were: 3x2 and 1/ (3 root3of(y2))

When I multiplied them together I got: x2 / root3of(y2)). Where I got stuck.

 

Can you show me the correct answer? 

Thanks.

 Jul 26, 2017
edited by Guest  Jul 26, 2017

Best Answer 

 #1
avatar+27578 
+3

As follows:

 

\(y=x^3\quad \frac{dy}{dx}=3x^2\\x=y^{1/3}\quad \frac{dx}{dy}=\frac{1}{3}y^{-2/3}\rightarrow \frac{1}{3y^{2/3}} \rightarrow \frac{1}{3(y^{1/3})^2}\rightarrow \frac{1}{3x^2}\)

 

Now you should be able to see that dy/dx*dx/dy = 1

.

 Jul 26, 2017
 #1
avatar+27578 
+3
Best Answer

As follows:

 

\(y=x^3\quad \frac{dy}{dx}=3x^2\\x=y^{1/3}\quad \frac{dx}{dy}=\frac{1}{3}y^{-2/3}\rightarrow \frac{1}{3y^{2/3}} \rightarrow \frac{1}{3(y^{1/3})^2}\rightarrow \frac{1}{3x^2}\)

 

Now you should be able to see that dy/dx*dx/dy = 1

.

Alan Jul 26, 2017
 #2
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-1

                           x+y

 Jul 27, 2017

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