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

Problem

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

Solution

1. Identify the function as the tangent function, which can be expressed as the ratio of the sine and cosine functions.

tan(x)=sin(x)/cos(x)

2. Apply the quotient rule for differentiation, which states that for a function u/v the derivative is (vd(u)/d(x)−ud(v)/d(x))/(v2)

d(tan(x))/d(x)=(cos(x)d(sin(x))/d(x)−sin(x)d(cos(x))/d(x))/cos2(x)

3. Substitute the known derivatives d(sin(x))/d(x)=cos(x) and d(cos(x))/d(x)=−sin(x) into the expression.

d(tan(x))/d(x)=(cos(x)*cos(x)−sin(x)*(−sin(x)))/cos2(x)

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

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

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

5. Rewrite the result using the reciprocal trigonometric identity sec(x)=1/cos(x)

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

Final Answer

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

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