Find the 2nd Derivative sec(x)^2

Problem

d2(sec2(x))/(d(x)2)

Solution

1. Identify the function as ƒ(x)=sec2(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(sec2(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(sec2(x))/d(x)=2*sec(x)⋅sec(x)*tan(x)

4. Simplify the first derivative expression.

d(sec2(x))/d(x)=2*sec2(x)*tan(x)

5. Apply the product rule to find the second derivative, where the two parts are 2*sec2(x) and tan(x)

d2(sec2(x))/(d(x)2)=(d(2)*sec2(x))/d(x)⋅tan(x)+2*sec2(x)⋅d(tan(x))/d(x)

6. Differentiate each part. We already know the derivative of 2*sec2(x) is 4*sec2(x)*tan(x) and the derivative of tan(x) is sec2(x)

d2(sec2(x))/(d(x)2)=(4*sec2(x)*tan(x))⋅tan(x)+2*sec2(x)⋅sec2(x)

7. Simplify the resulting expression by combining terms.

d2(sec2(x))/(d(x)2)=4*sec2(x)*tan2(x)+2*sec4(x)

Final Answer

d2(sec2(x))/(d(x)2)=4*sec2(x)*tan2(x)+2*sec4(x)

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