Find the 2nd Derivative -x/y

Problem

d2()/(d(x)2)*(−x/y)

Solution

1. Identify the expression as a function where y is implicitly a function of x and let z=−x/y

2. Apply the quotient rule to find the first derivative with respect to x

d(z)/d(x)=−(y(1)−xd(y)/d(x))/(y2)

3. Simplify the first derivative expression.

d(z)/d(x)=(xd(y)/d(x)−y)/(y2)

4. Apply the quotient rule again to find the second derivative.

d2(z)/(d(x)2)=(y2d()/d(x)*(xd(y)/d(x)−y)−(xd(y)/d(x)−y)d(y2)/d(x))/((y2)2)

5. Differentiate the terms in the numerator using the product rule and chain rule.

d()/d(x)*(xd(y)/d(x)−y)=d(y)/d(x)+xd2(y)/(d(x)2)−d(y)/d(x)=xd2(y)/(d(x)2)

d(y2)/d(x)=2*yd(y)/d(x)

6. Substitute these results back into the second derivative formula.

d2(z)/(d(x)2)=(y2*(xd2(y)/(d(x)2))−(xd(y)/d(x)−y)*(2*yd(y)/d(x)))/(y4)

7. Simplify the expression by factoring out y from the numerator and denominator.

d2(z)/(d(x)2)=(x*yd2(y)/(d(x)2)−2*x*(d(y)/d(x))2+2*yd(y)/d(x))/(y3)

Final Answer

d2()/(d(x)2)*(−x/y)=(x*yd2(y)/(d(x)2)−2*x*(d(y)/d(x))2+2*yd(y)/d(x))/(y3)

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