How to Simplify the Given Expression | Expand Numerator and Simplify Denominator

y’=((x²+3x+5)(cosx)-sinx(2x+3))/(x²+3x+5)²

To simplify the given expression, we can start by expanding and simplifying the numerator and denominator separately

To simplify the given expression, we can start by expanding and simplifying the numerator and denominator separately.

Numerator:
The numerator consists of two terms, (x²+3x+5)(cosx) and -sinx(2x+3). Let’s expand each term individually:

Term 1: (x²+3x+5)(cosx)
To expand this using the distributive property, multiply each term in (x²+3x+5) by cosx:

(x²+3x+5)(cosx) = x²cosx + 3xcosx + 5cosx

Term 2: -sinx(2x+3)
Using the distributive property, multiply -sinx by each term in (2x+3):

-sinx(2x+3) = -2xsinx – 3sinx

Now, let’s rewrite the numerator with the expanded terms:

Numerator: (x²cosx + 3xcosx + 5cosx) + (-2xsinx – 3sinx)
Numerator: x²cosx + 3xcosx + 5cosx – 2xsinx – 3sinx

Denominator:
The denominator is simply (x²+3x+5)², which is already in its simplified form.

Now, let’s rewrite the whole expression with the expanded numerator and simplified denominator:

y’ = (x²cosx + 3xcosx + 5cosx – 2xsinx – 3sinx)/(x²+3x+5)²

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