Find dy/dx y=tan(x)

Problem

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

Solution

1. Rewrite the tangent function in terms of sine and cosine using the quotient identity.

y=sin(x)/cos(x)

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

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

3. Differentiate the sine and cosine functions.

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

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

5. Use the reciprocal identity to express the final result in terms of the secant function.

d(y)/d(x)=sec2(x)

Final Answer

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

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