In this problem, we're asked to find the slope of the tangent line of our function f(x) at \( x = \frac{\pi}{4} \). The function that we're given here is \( f(x) = \sec(x) \). So, how do we find the slope of the tangent line? Remember, the slope of our tangent line is just the derivative. So, we want to go ahead and find the derivative of our function, \( f'(x) \).
Now, the derivative of our function, \( \sec(x) \), if we take this derivative, that's just going to give us \( \tan(x) \times \sec(x) \) based on what we know about the derivatives of trigonometric functions. But remember, this specifically asks us to find the slope of the tangent line at a value \( x = \frac{\pi}{4} \). That means we need to go ahead and plug that value into our derivative. So, here we want to find \( f'(\frac{\pi}{4}) \), and we get that by plugging \( \frac{\pi}{4} \) in. So, here we have \( \tan(\frac{\pi}{4}) \times \sec(\frac{\pi}{4}) \).
Now, \( \tan(\frac{\pi}{4}) \) is just equal to 1, and that's multiplying \( \sec(\frac{\pi}{4}) \), which based on our knowledge of trigonometric functions works out to be just the square root of 2. Now, one times the square root of 2 is just the square root of 2. So, this is the slope of the tangent line of our function at \( x = \frac{\pi}{4} \). Remember that everything that we've learned about derivatives still applies to these functions even when they involve trigonometric functions. Let me know if you have any questions here, and I'll see you in the next video.
