Question

dada a funçao f(x)=x^2-x+5. calcule {f(x+delta x)- f(x)}/deltax

Answer 1

$f(x)=x^{2}-x+5\\ f(x+\Delta x)=(x+\Delta x)^{2}-(x+\Delta x)+5$
$\frac{f(x+\Delta x)-f(x)}{\Delta x}=\\ =\frac{(x+\Delta x)^{2}-(x+\Delta x)+5-(x^{2}-x+5)}{\Delta x}\\ =\frac{x^{2}+2x\Delta x+(\Delta x)^{2}-x-\Delta x+5-x^{2}+x-5}{\Delta x}\\ =\frac{2x\Delta x+(\Delta x)^{2}-\Delta x}{\Delta x}\\ =\frac{\Delta x(2x+\Delta x-1)}{\Delta x}\\ =\boxed{\boxed{2x+\Delta x-1}}$