Sure, let's work through this problem step-by-step. We are asked to simplify the expression:
[tex]\[
\frac{\sin(x)}{e^{x^2}}
\][/tex]
1. Understand the components of the function:
- [tex]\(\sin(x)\)[/tex]: This is the sine function, a common trigonometric function.
- [tex]\(e^{x^2}\)[/tex]: This represents the exponential function with the exponent [tex]\(x^2\)[/tex].
2. Combine the functions:
- We want to express [tex]\(\frac{\sin(x)}{e^{x^2}}\)[/tex] in a different form that may be simpler or more insightful.
3. Rewriting the exponential function:
- By convention, we know that [tex]\(e^{x^2}\)[/tex] is the same as [tex]\(\exp(x^2)\)[/tex].
- A helpful property of exponents is that [tex]\(e^{-a} = \frac{1}{e^a}\)[/tex]. Applying this property in reverse, we can write:
[tex]\[
\frac{\sin(x)}{e^{x^2}} = \sin(x) \cdot e^{-x^2}
\][/tex]
4. Combine the functions into one expression:
- Notice that this is now a product of two functions, which might be easier to handle in various mathematical contexts:
[tex]\[
\sin(x) \cdot e^{-x^2}
\][/tex]
Therefore, the simplified form of the given function is:
[tex]\[
\boxed{e^{-x^2} \sin(x)}
\][/tex]
This is the final expression where we have rewritten [tex]\(\frac{\sin(x)}{e^{x^2}}\)[/tex] using properties of exponents to get a product of [tex]\(\sin(x)\)[/tex] and [tex]\(e^{-x^2}\)[/tex].
