Question

Find the first and second derivative for f(x) = x^2 - 9 / X^2 -4

please show me how you got the answer for both

Answer 1

f = (x^2-9)/(x^2-4)
f = u/v, so f' = (u'v-uv')/v^2
f' = ((2x)(x^2-4) - (x^2-9)(2x))/(x^2-4)^2
= 10x/(x^2-4)^2

or, realize that f(x) = 1 - 5/(x^2-4)
f' = 10x/(x^2-4)^2

f'' = ((10)(x^2-4)^2 - (10x)(2)(x^2-4)(2x))/(x^2-4)^4
= (10(x^2-4) - 40x^2)/(x^2-4)^3
= -10(3x^2+4)/(x^2-4)^3

Answer 2

2x+18x^-3

Answer 3

Use the quotient rule. (DN'-D'N)/(D)^2
D=denominator D'=derivative of denominator
N=numerator N'=derivative of numerator