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Calculating Derivative – Proof of an equation with derivatives – Exercise 6284


Given the following function


Prove that the following holds:

2y'^2-y\cdot y''=2y^3


First, we compute the first derivative and the second derivative, since they appear in the equation that needs to be proved.


Using Derivative formulas and the quotient rule in Derivative Rules, we get the derivative:


We want to compute the second derivative. To do this, we derive the first derivative and get:

y''=\frac{-2{(x^2-1)}^2+2x\cdot 2(x^2-1)\cdot 2x}{{(x^2-1)}^4}=

We simplify the second derivative:

=\frac{-2{(x^2-1)}^2+8x^2\cdot (x^2-1)}{{(x^2-1)}^4}=




We set the function and the derivative on the left side of the equation we need to prove, and we want to get the expression 0n the right side .

2y'^2-y\cdot y''=

=2\cdot {(\frac{-2x}{{(x^2-1)}^2})}^2-\frac{1}{x^2-1}\cdot \frac{6x^2+2}{{(x^2-1)}^3}=







We were able to reach the right side of the equation.

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