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

Problem

d(sec(x))/d(x)

Solution

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

2. Rewrite the function using the reciprocal identity sec(x)=1/cos(x) to prepare for the quotient rule or chain rule.

3. Apply the chain rule or the reciprocal rule, noting that the derivative of cos(x) is −sin(x)

4. Differentiate the expression:

d(sec(x))/d(x)=d(cos(x))/d(x)

5. Simplify the result using the power rule and chain rule:

−1*(cos(x))(−2)⋅(−sin(x))

6. Rearrange the terms into trigonometric identities:

sin(x)/cos2(x)=1/cos(x)⋅sin(x)/cos(x)

7. Substitute the identities sec(x)=1/cos(x) and tan(x)=sin(x)/cos(x)

Final Answer

d(sec(x))/d(x)=sec(x)*tan(x)

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