To find the expression for [tex]\( y \)[/tex], let's break down the given function and understand its components step-by-step.
We have the function:
[tex]\[ y = \frac{4 \sin(5x)}{5x^4} \][/tex]
Here is the step-by-step solution:
1. Numerator Analysis:
- The numerator of the fraction is [tex]\(4 \sin(5x)\)[/tex].
- Here, we have a trigonometric function [tex]\(\sin(5x)\)[/tex], which means the sine of the angle [tex]\(5x\)[/tex]. This is multiplied by the constant coefficient 4.
2. Denominator Analysis:
- The denominator of the fraction is [tex]\(5x^4\)[/tex].
- This includes the constant coefficient 5 and the variable term [tex]\(x^4\)[/tex], which means [tex]\(x\)[/tex] raised to the power of 4.
3. Combining Both Parts:
- The function as a whole, [tex]\( y = \frac{4 \sin(5x)}{5x^4} \)[/tex], is a ratio of the trigonometric expression in the numerator to the polynomial expression in the denominator.
These steps bring us to the complete function:
[tex]\[ y = \frac{4 \sin(5x)}{5 x^4} \][/tex]
As a result, the expression for [tex]\( y \)[/tex] is:
[tex]\[ y = \frac{4 \sin(5x)}{5 x^4} \][/tex]
This is a simplified form of the given function, and there are no further simplifications or factorings needed for this particular expression.
