Question

If f(x)=(x^2- 1),/(x^2 + 1), determine f'(x) and f"(x). Verificate whether your answers in (a) are reasonable by comparing the graphs of, f' and f".

Answer 1

`f'(x) = (4x)/((x^2 + 1)^2)\\\\f''(x) = (4(-3x^2 + 1))/((x^2 + 1)^3)`

Explanation:

If given:

`h(x) = f(x) \cdot g(x)`

then:

`h'(x) = (f'(x) \cdot g(x) - f(x) \cdot g(x))/((g(x))^2)`

Knowing this rule, we can use it to find the two derivatives:

`f(x) = (x^2 - 1)/(x^2 + 1)\\\\\to f'(x) = (2x(x^2 + 1)-2x(x^2 - 1))/((x^2 + 1)^2)\\\\= (2x^3 + 2x - 2x^2 + 2x)/((x^2 + 1)^2)\\\\f'(x) = (4x)/((x^2 + 1)^2)\\\\\to f''(x) = (4(x^2 + 1)^2 - 2(x^2 + 1) * 2x * 4x)/((x^2 + 1)^4)\\\\= (4(x^2 + 1)^2 - 16x^2(x^2 + 1))/((x^2 + 1)^4)\\\\= (4(x^2 + 1)\left[(x^2 + 1) - 4x^2\right])/((x^2 + 1)^4)\\\\f''(x) = (4(-3x^2 + 1))/((x^2 + 1)^3)`