The provided image shows a mathematical expression:

1. Expression: $y = (x^2 - 6x)^{-\frac{7}{2}}$
2. Task: Compute the derivative $\frac{dy}{dx}$.

Let’s compute it step-by-step:

Step 1: Use the Chain Rule

We will differentiate $y = u^{-\frac{7}{2}}$, where $u = x^2 - 6x$.

$\frac{dy}{dx} = \frac{d}{du} \left( u^{-\frac{7}{2}} \right) \cdot \frac{du}{dx}$

The derivative of $u^{-\frac{7}{2}}$ is:

$\frac{d}{du} \left( u^{-\frac{7}{2}} \right) = -\frac{7}{2} u^{-\frac{9}{2}}$

Step 2: Compute $\frac{du}{dx}$

Here, $u = x^2 - 6x$, so:

$\frac{du}{dx} = 2x - 6$

Step 3: Combine Results

Now substitute back into the chain rule:

$\frac{dy}{dx} = -\frac{7}{2} (x^2 - 6x)^{-\frac{9}{2}} \cdot (2x - 6)$

Final Answer:

$\frac{dy}{dx} = -\frac{7 (2x - 6)}{2 (x^2 - 6x)^{\frac{9}{2}}}$

Would you like further clarification or details on any specific part of the process?

Questions for deeper understanding:

1. What are some common applications of chain rule in calculus?
2. How can you verify the derivative result is correct?
3. How does changing the exponent in ( u