Use logarithmic differentiation to prove D9: d/dx (1/u^n) =( -n/u^n+1)(du/dx)
f(x)=1unf(x)=\frac{1}{u^n}f(x)=un1
lnf(x)=ln1un=−nlnu\ln{f(x)}=\ln{\frac{1}{u^n}}=-n\ln ulnf(x)=lnun1=−nlnu
ddxlnf(x)=ddx(−nlnu)\frac{d}{dx}\ln{f(x)}=\frac{d}{dx}(-n\ln u)dxdlnf(x)=dxd(−nlnu)
1f(x)df(x)dx=−n1ududx\frac{1}{f(x)}\frac{df(x)}{dx}=-n\frac{1}{u}\frac{du}{dx}f(x)1dxdf(x)=−nu1dxdu
df(x)dx=−nf(x)1ududx=−n1un+1dudx\frac{df(x)}{dx}=-nf(x)\frac{1}{u}\frac{du}{dx}=-n\frac{1}{u^{n+1}}\frac{du}{dx}dxdf(x)=−nf(x)u1dxdu=−nun+11dxdu
ddx1un=−n1un+1dudx\frac{d}{dx}\frac{1}{u^n}=-n\frac{1}{u^{n+1}}\frac{du}{dx}dxdun1=−nun+11dxdu
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