Find dy/dx y=(sin(x))/(cos(x))

Problem

d()/d(x)sin(x)/cos(x)

Solution

1. Identify the function as a quotient of two trigonometric functions, where u=sin(x) and v=cos(x)

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

3. Differentiate the numerator and denominator: d(sin(x))/d(x)=cos(x) and d(cos(x))/d(x)=−sin(x)

4. Substitute these derivatives into the quotient rule formula:

d(y)/d(x)=(cos(x)*cos(x)−sin(x)*(−sin(x)))/((cos(x))2)

5. Simplify the numerator using the Pythagorean identity sin2(x)+cos2(x)=1

d(y)/d(x)=(cos2(x)+sin2(x))/cos2(x)

d(y)/d(x)=1/cos2(x)

6. Apply the reciprocal identity sec(x)=1/cos(x) to reach the final form.

Final Answer

d()/d(x)sin(x)/cos(x)=sec2(x)

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