How To: Use implicit differentiation

Use implicit differentiation

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=0-18x, 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. =-9y-81x²/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/1-81x²/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/1-81x² /y times 1/y² you can reduce the fraction to -9y/1-81x² / 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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