Find dy/dx y=x/(2-tan(x))

Problem

d()/d(x)x/(2−tan(x))

Solution

1. Identify the rule needed for differentiation. Since the function is a quotient of two expressions, apply the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Assign the variables for the numerator and denominator. Let u=x and v=2−tan(x)

3. Differentiate the individual components.

d(x)/d(x)=1

d(2−tan(x))/d(x)=−sec2(x)

4. Substitute these derivatives into the quotient rule formula.

d(y)/d(x)=((2−tan(x))*(1)−(x)*(−sec2(x)))/((2−tan(x))2)

5. Simplify the numerator by distributing the terms and combining signs.

d(y)/d(x)=(2−tan(x)+x*sec2(x))/((2−tan(x))2)

Final Answer

d(y)/d(x)=(2−tan(x)+x*sec2(x))/((2−tan(x))2)

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