Sure, let's find the derivative of the function [tex]\( f(x) = \frac{5 - x^2}{8 + x^2} \)[/tex].
To find [tex]\( f'(x) \)[/tex], we use the quotient rule for derivatives. The quotient rule states that if you have a function [tex]\( f(x) = \frac{g(x)}{h(x)} \)[/tex], then the derivative [tex]\( f'(x) \)[/tex] is given by:
[tex]\[
f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}
\][/tex]
In our case, [tex]\( g(x) = 5 - x^2 \)[/tex] and [tex]\( h(x) = 8 + x^2 \)[/tex].
1. First, compute the derivatives of [tex]\( g(x) \)[/tex] and [tex]\( h(x) \)[/tex]:
[tex]\[
g(x) = 5 - x^2 \quad \Rightarrow \quad g'(x) = -2x
\][/tex]
[tex]\[
h(x) = 8 + x^2 \quad \Rightarrow \quad h'(x) = 2x
\][/tex]
2. Then, substitute into the quotient rule formula:
[tex]\[
f'(x) = \frac{(5 - x^2)'(8 + x^2) - (5 - x^2)(8 + x^2)'}{(8 + x^2)^2}
\][/tex]
3. Calculate the numerator:
[tex]\[
\text{Numerator} = g'(x)h(x) - g(x)h'(x) = (-2x)(8 + x^2) - (5 - x^2)(2x)
\][/tex]
4. Simplify the expression inside the numerator:
[tex]\[
(-2x)(8 + x^2) = -16x - 2x^3
\][/tex]
[tex]\[
(5 - x^2)(2x) = 10x - 2x^3
\][/tex]
5. Combine these results:
[tex]\[
\text{Numerator} = -16x - 2x^3 - (10x - 2x^3)
\][/tex]
[tex]\[
= -16x - 2x^3 - 10x + 2x^3
\][/tex]
[tex]\[
= -16x - 10x
\][/tex]
[tex]\[
= -26x
\][/tex]
6. Finally, divide by the square of the denominator:
[tex]\[
f'(x) = \frac{-26x}{(8 + x^2)^2}
\][/tex]
So, the derivative of the function [tex]\( f(x) = \frac{5 - x^2}{8 + x^2} \)[/tex] is:
[tex]\[
f'(x) = -\frac{2x(5 - x^2)}{(8 + x^2)^{2}} - \frac{2x}{(8 + x^2)}
\][/tex]
