Find the 2nd Derivative f(x)=sec(x)

Problem

ƒ(x)=sec(x)

Solution

1. Identify the function ƒ(x)=sec(x) and the goal to find the second derivative ƒ(x)″

2. Find the first derivative by applying the standard trigonometric differentiation rule for the secant function.

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

3. Apply the product rule to find the second derivative, where the product rule is (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

ƒ(x)″=sec(x)d(tan(x))/d(x)+tan(x)d(sec(x))/d(x)

4. Differentiate the individual components tan(x) and sec(x)

ƒ(x)″=sec(x)*(sec2(x))+tan(x)*(sec(x)*tan(x))

5. Simplify the expression by multiplying the terms and factoring out common factors if necessary.

ƒ(x)″=sec3(x)+sec(x)*tan2(x)

6. Factor the expression to reach the final form.

ƒ(x)″=sec(x)*(sec2(x)+tan2(x))

Final Answer

ƒ(x)″=sec(x)*(sec2(x)+tan2(x))

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