Find the Derivative - d/dx -x/y

Problem

d()/d(x)*(−x/y)

Solution

1. Identify the rule needed for differentiation. Since the expression is a quotient of two variables where y is an implicit function of x use the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Assign the variables for the quotient rule. Let u=−x and v=y

3. Differentiate the numerator and denominator with respect to x The derivative of the numerator is (d(−)*x)/d(x)=−1 The derivative of the denominator is d(y)/d(x)=y′

4. Apply the quotient rule formula by substituting the assigned variables and their derivatives.

(y*(−1)−(−x)*y′)/(y2)

5. Simplify the resulting expression by performing the multiplication and combining signs.

(−y+x*y′)/(y2)

Final Answer

d()/d(x)*(−x/y)=(x*y′−y)/(y2)

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