To find the derivative [tex]\( g'(x) \)[/tex] of the function [tex]\( g(x) = \frac{4x^2 - 9}{4x + 6} \)[/tex], we will use the quotient rule for differentiation, which is given by:
[tex]\[ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \][/tex]
Here, [tex]\( u(x) = 4x^2 - 9 \)[/tex] and [tex]\( v(x) = 4x + 6 \)[/tex]. We need to find the derivatives of [tex]\( u \)[/tex] and [tex]\( v \)[/tex]:
1. Differentiate [tex]\( u(x) = 4x^2 - 9 \)[/tex]:
[tex]\[ u'(x) = \frac{d}{dx}[4x^2 - 9] = 8x \][/tex]
2. Differentiate [tex]\( v(x) = 4x + 6 \)[/tex]:
[tex]\[ v'(x) = \frac{d}{dx}[4x + 6] = 4 \][/tex]
Now, we apply the quotient rule:
[tex]\[ g'(x) = \frac{(4x + 6)(8x) - (4x^2 - 9)(4)}{(4x + 6)^2} \][/tex]
We simplify the numerator step by step:
[tex]\[ (4x + 6)(8x) = 32x^2 + 48x \][/tex]
[tex]\[ (4x^2 - 9)(4) = 16x^2 - 36 \][/tex]
Subtract the second term from the first:
[tex]\[ 32x^2 + 48x - (16x^2 - 36) = 32x^2 + 48x - 16x^2 + 36 \][/tex]
Combine like terms:
[tex]\[ 32x^2 - 16x^2 + 48x + 36 = 16x^2 + 48x + 36 \][/tex]
Thus, the numerator simplifies to:
[tex]\[ 8x(4x + 6) - 4(4x^2 - 9) \][/tex]
Combine both parts:
[tex]\[ g'(x) = \frac{8x(4x + 6) - 4(4x^2 - 9)}{(4x + 6)^2} \][/tex]
Recognizing that the numerator is a combination of simplified polynomial terms, we have:
[tex]\[ g'(x) = \frac{8x}{(4x + 6)} - \frac{4(4x^2 - 9)}{(4x + 6)^2} \][/tex]
Therefore, the derivative [tex]\( g'(x) \)[/tex] is:
[tex]\[ g'(x) = \frac{8x}{4x + 6} - \frac{4(4x^2 - 9)}{(4x + 6)^2} \][/tex]
This gives us the final, simplified expression for the derivative of the given function [tex]\( g(x) \)[/tex].
