Use the equation given below to find f '' ( 𝜋 /3) .

Question

f(X)=sec(x)

Answer 1

You have to start by taking the second derivative.

f(x)=sec(x)

f'(x) = sec(x)tan(x) (this should be memorized. The proof isn't bad, but it takes multiplying by an unusual term)

To find the second derivative, you need to remember to use the product rule. d/dx[sec(x)] = sec(x)tan(x) and d/dx[tan(x)] = sec²(x).

f''(x) = [sec(x)tan(x)]•tan(x) + sec(x)•[sec²(x)] (product rule expanded form)

After we get the simplified form f''(x) = sec(x)tan²(x) + sec³(x), we need only to plug π/3 in for x.

f''(π/3) = sec(π/3)tan²(π/3) + sec³(π/3)

f''(π/3) = 2•√3² + 2³

f''(π/3) = 14