Sure, I'll guide you through how to derive the expression [tex]\(\sin x \sqrt{x^2+1}\)[/tex].
1. Understand the Function:
The given function is [tex]\( f(x) = \sin x \cdot \sqrt{x^2 + 1} \)[/tex].
This is a product of two parts:
- Trigonometric function: [tex]\(\sin x\)[/tex]
- Radical function: [tex]\(\sqrt{x^2 + 1}\)[/tex]
2. Components Breakdown:
- [tex]\(\sin x\)[/tex]: This is a standard sine function which oscillates between -1 and 1.
- [tex]\(\sqrt{x^2 + 1}\)[/tex]: This represents the square root of the sum of [tex]\(x^2\)[/tex] and 1. It's always positive since the argument inside the square root is always positive for all real [tex]\(x\)[/tex].
3. Combine the Functions:
To combine these two parts, you multiply them together:
[tex]\[
f(x) = \sin x \cdot \sqrt{x^2 + 1}
\][/tex]
4. Express the Functions Together:
Thus, the expression for [tex]\(f(x)\)[/tex] is indeed:
[tex]\[
f(x) = \sin x \sqrt{x^2 + 1}
\][/tex]
There you have it—a detailed breakdown of the function [tex]\(\sin x \sqrt{x^2 + 1}\)[/tex].
