Find the Derivative - d/dx (sin(x))/(cos(x))

Problem

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

Solution

1. Identify the rule needed for the derivative of a quotient of two functions, which is the quotient rule.

2. Apply the formula for the quotient rule, where the derivative of u/v is (vd(u)/d(x)−ud(v)/d(x))/(v2)

3. Substitute u=sin(x) and v=cos(x) into the formula.

4. Calculate the individual derivatives: d(sin(x))/d(x)=cos(x) and d(cos(x))/d(x)=−sin(x)

5. Plug in these derivatives into the quotient rule expression.

(cos(x)*cos(x)−sin(x)*(−sin(x)))/cos2(x)

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

(cos2(x)+sin2(x))/cos2(x)

7. Rewrite the expression using the trigonometric identity sec(x)=1/cos(x)

1/cos2(x)

Final Answer

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

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