Find the Derivative - d/dx 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 secant 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 1/u is −1/(u2)⋅d(u)/d(x)

4. Differentiate the inner function cos(x) which results in −sin(x)

5. Substitute the derivative back into the expression:

d(sec(x))/d(x)=−1/cos2(x)⋅(−sin(x))

6. Simplify the resulting expression by separating the terms:

d(sec(x))/d(x)=sin(x)/cos2(x)

7. Factor the expression into basic trigonometric functions:

d(sec(x))/d(x)=1/cos(x)⋅sin(x)/cos(x)

8. Substitute the identities sec(x)=1/cos(x) and tan(x)=sin(x)/cos(x) to reach the final form.

Final Answer

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

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