Let’s break down and solve the expression step by step.

Your query can be interpreted as:

$\frac{d}{dx} \left( \frac{2}{3} \int_0^x e^{-t^2} dt \right)$

Solution:

1. Simplify the expression inside the derivative:

   • The constant $\frac{2}{3}$ can be factored out of the integral: $\frac{2}{3} \int_0^x e^{-t^2} dt$

2. Differentiate the expression with respect to $x$:

   By the Fundamental Theorem of Calculus, the derivative of $\int_0^x f(t) \, dt$ is $f(x)$, so: $\frac{d}{dx} \left( \frac{2}{3} \int_0^x e^{-t^2} dt \right) = \frac{2}{3} e^{-x^2}.$

Final Answer:

$\frac{2}{3} e^{-x^2}$

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Do you want further details on the steps or explanations for this? Here are some related questions to explore:

1. What is the Fundamental Theorem of Calculus?
2. How does differentiation and integration relate to each other?
3. Can we evaluate $\int_0^x e^{-t^2} dt$ explicitly?
4. What are common applications of Gaussian functions ($e^{-x^2}$) in mathematics?
5. How do constants like $\frac{2}{3}$ impact the differentiation of integrals?

Tip: Always check if you can factor out constants before applying calculus rules; it simplifies computations!