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

Problem

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

Solution

1. Identify the rule needed for differentiation. Since the function is a quotient of two trigonometric functions, 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=sin(x) and v=cos(x)

3. Differentiate the individual components. The derivative of the numerator is d(sin(x))/d(x)=cos(x) and the derivative of the denominator is d(cos(x))/d(x)=−sin(x)

4. Substitute these values into the quotient rule formula.

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

5. Simplify the numerator using the distributive property.

d()/d(x)sin(x)/cos(x)=(cos2(x)+sin2(x))/cos2(x)

6. Apply the Pythagorean identity sin2(x)+cos2(x)=1 to the numerator.

d()/d(x)sin(x)/cos(x)=1/cos2(x)

7. Use the reciprocal identity sec(x)=1/cos(x) to write the final result.

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

Final Answer

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

Want more problems? Check here!