To differentiate $y = 5^{x^2 - 1}$ with respect to $x$, we can use logarithmic differentiation.

1. Start with the original function: $y = 5^{x^2 - 1}$

2. Rewrite the expression using the exponential function, $a^u = e^{u \ln a}$: $y = e^{(x^2 - 1) \ln 5}$

3. Differentiate both sides with respect to $x$: $\frac{dy}{dx} = e^{(x^2 - 1) \ln 5} \cdot \frac{d}{dx} \left( (x^2 - 1) \ln 5 \right)$

4. Differentiate the exponent: $\frac{d}{dx} \left( (x^2 - 1) \ln 5 \right) = \ln 5 \cdot \frac{d}{dx} (x^2 - 1) = \ln 5 \cdot 2x$

5. Substitute back: $\frac{dy}{dx} = 5^{x^2 - 1} \cdot (2x \ln 5)$

So, the derivative is: $\frac{dy}{dx} = 5^{x^2 - 1} \cdot 2x \ln 5$

Would you like more details on any steps?

Here are some related questions:

1. How would you differentiate $y = a^{g(x)}$ in general?
2. How do you apply the chain rule in differentiating exponential functions?
3. What are some common logarithmic differentiation examples?
4. How does $\ln a$ affect derivatives of functions like $a^{f(x)}$?
5. How can you differentiate $y = e^{x^2 - 1}$?

Tip: For exponential functions with non-$e$ bases, rewriting in terms of $e$ often simplifies differentiation.