Hvordan deriverer man f(x)=x^2 + 1.5x ved hjelp av
(f(x)-f(x+deltax)-f(X))/deltax
Derivasjon
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Den deriverte av f(x) = x[sup]2[/sup] + 1,5x blir
lim_(Δx -> 0) [ (f(x + Δx) - f(x)) / Δx ]
= lim_(Δx -> 0) [(x + Δx)[sup]2[/sup] + 1,5(x + Δx) - (x[sup]2[/sup] + 1,5x) / Δx]
= lim_(Δx -> 0) [ (x[sup]2[/sup] + 2xΔx + Δx[sup]2[/sup] + 1,5x + 1,5Δx - x[sup]2[/sup] - 1,5x) / Δx ]
= lim_(Δx -> 0) [ (Δx[sup]2[/sup] + 2xΔx + 1,5Δx) / Δx ]
= lim_(Δx -> 0) [ Δx + 2x + 1,5 ]
= 2x + 1,5.
lim_(Δx -> 0) [ (f(x + Δx) - f(x)) / Δx ]
= lim_(Δx -> 0) [(x + Δx)[sup]2[/sup] + 1,5(x + Δx) - (x[sup]2[/sup] + 1,5x) / Δx]
= lim_(Δx -> 0) [ (x[sup]2[/sup] + 2xΔx + Δx[sup]2[/sup] + 1,5x + 1,5Δx - x[sup]2[/sup] - 1,5x) / Δx ]
= lim_(Δx -> 0) [ (Δx[sup]2[/sup] + 2xΔx + 1,5Δx) / Δx ]
= lim_(Δx -> 0) [ Δx + 2x + 1,5 ]
= 2x + 1,5.