Find the Derivative - d/dx (sec(x))/(1+sec(x))

Problem

d()/d(x)sec(x)/(1+sec(x))

Solution

1. Identify the rule needed for differentiation. Since the expression is a fraction of two functions, apply the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Assign the numerator and denominator functions. Let u=sec(x) and v=1+sec(x)

3. Differentiate both parts. The derivative of sec(x) is sec(x)*tan(x)

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

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

4. Substitute these into the quotient rule formula.

((1+sec(x))*(sec(x)*tan(x))−(sec(x))*(sec(x)*tan(x)))/((1+sec(x))2)

5. Distribute the terms in the numerator.

(sec(x)*tan(x)+sec2(x)*tan(x)−sec2(x)*tan(x))/((1+sec(x))2)

6. Simplify the numerator by canceling the terms sec2(x)*tan(x) and −sec2(x)*tan(x)

(sec(x)*tan(x))/((1+sec(x))2)

Final Answer

d()/d(x)sec(x)/(1+sec(x))=(sec(x)*tan(x))/((1+sec(x))2)

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