To differentiate $e^{n x^n}$ with respect to $x$, we use the chain rule. Here's the step-by-step process:

1. Identify the outer and inner functions:

   • Outer function: $e^u$, where $u = n x^n$.
   • Inner function: $u = n x^n$.

2. Differentiate the outer function:

   • The derivative of $e^u$ with respect to $u$ is $e^u$.

3. Differentiate the inner function:

   • The derivative of $n x^n$ with respect to $x$ is: $\frac{d}{dx}(n x^n) = n \cdot n x^{n-1} = n^2 x^{n-1}.$

4. Combine using the chain rule:

   • Multiply the derivative of the outer function by the derivative of the inner function: $\frac{d}{dx}(e^{n x^n}) = e^{n x^n} \cdot n^2 x^{n-1}.$

Final Answer:

$\frac{d}{dx}(e^{n x^n}) = n^2 x^{n-1} e^{n x^n}.$

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Would you like a detailed explanation of the chain rule or a worked-out example?

Related Questions:

1. How do you differentiate $\ln(x^n)$?
2. How would the result change if $n$ is a function of $x$?
3. What happens if $x^n$ is replaced by $x^{n+1}$?
4. How do you integrate $e^{n x^n}$?
5. Can you explain the relationship between exponential and logarithmic differentiation?

Tip:

Always simplify inner derivatives fully before multiplying them with the outer derivative to reduce errors.