If f(x) = sec x, find f''(π/4).

f'(x) = secxtanx then use the product rule and figure out the derivative of that:

f''(x) = tanxsecxtanx + secx sec^2x and then it becomes tan^2x + sec^3x.
then you plug in pi/4 which is 45 degrees.

it is (1)(1.41) + 2.83 and the answer is 4.24

To find the second derivative of f(x) = sec x and evaluate it at x = π/4, we follow these steps:

Step 1: Find the first derivative of f(x).
The derivative of f(x) = sec x can be found using the chain rule. The chain rule states that if we have a composition of functions, where g(x) = sec x and h(x) = x, then the derivative of f(g(x)) is given by f'(g(x)) * g'(x). Applying the chain rule, we get:

f'(x) = sec x * tan x

Step 2: Find the second derivative of f(x).
To find the second derivative, we differentiate the first derivative obtained in step 1. Applying the product rule, we have:

f''(x) = (sec x * tan x)'
= (sec x * tan x)' + (sec x)' * tan x
= sec x * sec x + sec x * tan^2 x
= sec^2 x + sec x * tan^2 x

Step 3: Evaluate f''(π/4).
Now that we have the second derivative f''(x) = sec^2 x + sec x * tan^2 x, we substitute x = π/4:

f''(π/4) = sec^2 (π/4) + sec (π/4) * tan^2 (π/4)

Using the trigonometric identities sec^2 θ = 1 + tan^2 θ and tan θ = 1, we can simplify the expression:

f''(π/4) = (1 + tan^2 (π/4)) + sec (π/4) * 1
= 1 + 1 + sec (π/4)
= 2 + √2

Therefore, f''(π/4) = 2 + √2.