Find the Derivative Using Quotient Rule - d/dx -x/y

Problem

d()/d(x)*(−x/y)

Solution

1. Identify the function as a quotient where u=−x and v=y and note that y is a function of x

2. Apply the quotient rule formula, which is d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

3. Differentiate the numerator u=−x with respect to x to get (d(−)*x)/d(x)=−1

4. Differentiate the denominator v=y with respect to x using implicit differentiation to get d(y)/d(x)

5. Substitute these values into the quotient rule formula:

(y*(−1)−(−x)d(y)/d(x))/(y2)

6. Simplify the expression by distributing the signs in the numerator.

(−y+xd(y)/d(x))/(y2)

Final Answer

d()/d(x)*(−x/y)=(xd(y)/d(x)−y)/(y2)

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