Find the Derivative - d/d@VAR f(x)=tan(x)

Problem

d(tan(x))/d(x)

Solution

1. Identify the function to be differentiated, which is ƒ(x)=tan(x)

2. Express the tangent function in terms of sine and cosine using the quotient identity tan(x)=sin(x)/cos(x)

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

4. Substitute u=sin(x) and v=cos(x) into the quotient rule formula.

5. Differentiate the numerator components: d(sin(x))/d(x)=cos(x) and d(cos(x))/d(x)=−sin(x)

6. Simplify the resulting expression: (cos(x)*cos(x)−sin(x)*(−sin(x)))/cos2(x)

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

8. Simplify the fraction 1/cos2(x) using the reciprocal identity sec(x)=1/cos(x)

Final Answer

d(tan(x))/d(x)=sec2(x)

Want more problems? Check here!