Taking this formula 9x²+y²=9 your instructor shows you how to find the solution for the variable y. Finding a second derivative is no more difficult that finding the first derivative, you only have to do the problem equation through twice. The derivative of 9x² will be 18, of y² will be 2y [this is the same as 2(y), 2 x y or 2 * y]. The problem now goes 18x + 2y dy/dx=0. This equation is solved for dy/dx making the next step in the problem go 18x 18x+2y dy/dx=018x, 2y dy/dx= 18x. The next step 2y/2y dy/dx=18x/2y, dy/dx=9x/y (negative 9x divided by y). To get or find the derivative with respect to x, d2/y (second derivative with respect to y) /. Now, you need to take the bottom times the derivative of the top, the quotient with y like this y*(9)(9x)(1*dy/dx). The problem now reads d2/y / dx²=y*(9)(9x)(1*dy/dx)/y² the entire section after the equal sign to be divided by y². Replace the dy/dx with negative 9 over y as you found in the first step and reduce the problem. You now have 9/y=9y+9x(9/y) / y², the second side of the problem all being divided by y². Do some more reducing. =9y81x²/y / y², the entire side divided by y². Create a common denominator on the top fraction by placing 9y over 1 and multiplying. 9y/1 times 81x²/y / y² = 9y/181x²/y /y², the entire string being divided by y². The top and bottom of this fraction can be inverted and multiplied according to fraction multiplication rules. Given 9y/181x² /y times 1/y² you can reduce the fraction to 9y/181x² / y³. Factor out the negative 9 on the top resulting in 9(y²+9x²) / y³. You can replace 9x²+y² with 9 to solve the problem, this matches your beginning equation, and get 81/y³ for the second derivative in regards to x. These step by step instructions will simplify solving equations of this type.
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