Answer
a) 11.27
b) 0.48
Work Step by Step
a) $f(x) = x^{3}-x^{2}+1$
$f'(x) = 3x^{2}-2x$
$f'(x_{0}) = 3(x_{0})^{2} - 2(x_{0})$
As $x_{0} = 2.3$
$f'(2.3) = 3(2.3)^{2} - 2(2.3)$
After simplification :
$f'(2.3) = 11.27$
b) $f(x) = \frac{x}{x^{2}+1}$
$f'(x) = \frac{(x^{2}+1)(1)-x(2x)}{(x^{2}+1)^{2}}$
$f'(x) = \frac{1-x^{2}}{(x^{2}+1)^{2}}$
$f'(x_{0}) = \frac{1-(x_{0})^{2}}{((x_{0})^{2}+1)^{2}}$
As $x_{0} = -0.5$
$f'(-0.5) = \frac{1-(-0.5)^{2}}{((-0.5)^{2}+1)^{2}}$
After simplification :
$f'(-0.5) = 0.48$