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.

Discover how to differentiate the function y = 5^(x^2 - 1) using logarithmic differentiation and the chain rule. This solution provides step-by-step instructions for rewriting exponential functions and differentiating complex exponents, suitable for advanced high school students in Grades 11-12.

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