To find the derivative of the function $y = x^{7 \cos x}$ using logarithmic differentiation, we follow these steps:

1. Take the natural logarithm of both sides: $\ln y = \ln(x^{7 \cos x})$

2. Apply the properties of logarithms: $\ln y = 7 \cos x \cdot \ln x$

3. Differentiate both sides with respect to $x$: Using implicit differentiation: $\frac{d}{dx}(\ln y) = \frac{1}{y} \frac{dy}{dx}$ For the right side, we apply the product rule: $\frac{d}{dx}(7 \cos x \cdot \ln x) = 7(-\sin x \cdot \ln x) + 7 \cos x \cdot \frac{1}{x}$ So, $\frac{d}{dx}(7 \cos x \cdot \ln x) = 7(-\sin x \ln x + \frac{\cos x}{x})$

4. Combine the results: Putting it all together: $\frac{1}{y} \frac{dy}{dx} = 7(-\sin x \ln x + \frac{\cos x}{x})$ Therefore, $\frac{dy}{dx} = y \cdot 7(-\sin x \ln x + \frac{\cos x}{x})$

5. Substituting back for $y$: Since $y = x^{7 \cos x}$: $\frac{dy}{dx} = x^{7 \cos x} \cdot 7(-\sin x \ln x + \frac{\cos x}{x})$

Thus, the derivative is: $\frac{dy}{dx} = 7 x^{7 \cos x} \left(-\sin x \ln x + \frac{\cos x}{x}\right)$

Do you want further details or have any questions?

Here are some related questions you might find interesting:

1. What are the applications of logarithmic differentiation?
2. Can you explain the product rule and the chain rule in more detail?
3. How does logarithmic differentiation simplify the process for more complex functions?
4. What are the limits of using logarithmic differentiation?
5. Can you provide examples of functions where logarithmic differentiation is especially useful?

Tip: When using logarithmic differentiation, it’s particularly helpful for functions that involve variables raised to variable powers.