Find the 2nd Derivative sec(x)^2

Problem

d^2(sec^2(x))/(d(x)^2)

Solution

1. Identify the function as ƒ(x)=sec^2(x) and prepare to find the first derivative using the power rule and the chain rule.

2. Apply the chain rule to find the first derivative, treating the function as (sec(x))^2

d(sec^2(x))/d(x)=2*sec(x)⋅d(sec(x))/d(x)

3. Substitute the derivative of sec(x) which is sec(x)*tan(x)

d(sec^2(x))/d(x)=2*sec(x)⋅sec(x)*tan(x)

4. Simplify the first derivative expression.

d(sec^2(x))/d(x)=2*sec^2(x)*tan(x)

5. Apply the product rule to find the second derivative, where the two parts are 2*sec^2(x) and tan(x)

d^2(sec^2(x))/(d(x)^2)=(d(2)*sec^2(x))/d(x)⋅tan(x)+2*sec^2(x)⋅d(tan(x))/d(x)

6. Differentiate each part. We already know the derivative of 2*sec^2(x) is 4*sec^2(x)*tan(x) and the derivative of tan(x) is sec^2(x)

d^2(sec^2(x))/(d(x)^2)=(4*sec^2(x)*tan(x))⋅tan(x)+2*sec^2(x)⋅sec^2(x)

7. Simplify the resulting expression by combining terms.

d^2(sec^2(x))/(d(x)^2)=4*sec^2(x)*tan^2(x)+2*sec^4(x)

Final Answer

d^2(sec^2(x))/(d(x)^2)=4*sec^2(x)*tan^2(x)+2*sec^4(x)

Want more problems? Check here!