We are tasked with finding the derivative $f'(x)$ for the function:

$f(x) = \frac{\tan x - 5}{\sec x}$

Step 1: Use the Quotient Rule

To differentiate $f(x)$, we apply the quotient rule which states that if:

$f(x) = \frac{g(x)}{h(x)}$

then:

$f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}$

Here, $g(x) = \tan x - 5$ and $h(x) = \sec x$.

Step 2: Differentiate $g(x)$ and $h(x)$

• $g(x) = \tan x - 5$

  • $g'(x) = \sec^2 x$ because the derivative of $\tan x$ is $\sec^2 x$, and the derivative of a constant is 0.

• $h(x) = \sec x$

  • $h'(x) = \sec x \tan x$, since the derivative of $\sec x$ is $\sec x \tan x$.

Step 3: Apply the Quotient Rule

Now we plug everything into the quotient rule formula:

$f'(x) = \frac{(\sec^2 x)(\sec x) - (\tan x - 5)(\sec x \tan x)}{(\sec x)^2}$

Step 4: Simplify the Expression

First, expand the terms:

$f'(x) = \frac{\sec^3 x - (\tan^2 x - 5 \tan x)\sec x}{\sec^2 x}$

Now, simplify the fraction by canceling $\sec^2 x$:

$f'(x) = \frac{\sec^3 x}{\sec^2 x} - \frac{(\tan^2 x - 5 \tan x)\sec x}{\sec^2 x}$

$f'(x) = \sec x - \frac{\tan^2 x - 5 \tan x}{\sec x}$

$f'(x) = \sec x - (\tan^2 x - 5 \tan x) \cdot \cos x$

So the derivative is:

$f'(x) = \sec x - (\sin^2 x - 5 \sin x \cos x)$

Step 5: Find $f' \left( \frac{\pi}{4} \right)$

At $x = \frac{\pi}{4}$:

• $\sec \left( \frac{\pi}{4} \right) = \sqrt{2}$
• $\sin \left( \frac{\pi}{4} \right) = \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2}$

Substitute these values into the expression for $f'(x)$:

$f' \left( \frac{\pi}{4} \right) = \sqrt{2} - \left( \left( \frac{\sqrt{2}}{2} \right)^2 - 5 \times \frac{\sqrt{2}}{2} \times \frac{\sqrt{2}}{2} \right)$

$f' \left( \frac{\pi}{4} \right) = \sqrt{2} - \left( \frac{1}{2} - \frac{5}{2} \right)$

$f' \left( \frac{\pi}{4} \right) = \sqrt{2} - \left( \frac{-4}{2} \right)$

$f' \left( \frac{\pi}{4} \right) = \sqrt{2} + 2$

Thus, the value of the derivative at $x = \frac{\pi}{4}$ is:

$f' \left( \frac{\pi}{4} \right) = \sqrt{2} + 2$

Do you have any questions or would you like additional details?

Here are 5 related questions you might consider:

1. How do you apply the quotient rule to more complex expressions?
2. What is the geometric meaning of the derivative of $f(x)$?
3. How can we simplify trigonometric expressions involving both sine and cosine?
4. Why does the derivative of $\sec x$ involve both $\sec x$ and $\tan x$?
5. How do second derivatives change the shape of a function involving trigonometric terms?

Tip: Always simplify trigonometric derivatives before substituting numerical values—it reduces calculation complexity!