Find the Derivative - d/dx y=x/(2-tan(x))

Problem

d()/d(x)x/(2−tan(x))

Solution

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

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

3. Differentiate the individual components. The derivative of the numerator is d(x)/d(x)=1 The derivative of the denominator is d(2−tan(x))/d(x)=−sec2(x)

4. Substitute these values 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 and combining terms.

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

Final Answer

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

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