Pochodne

[thermaltake]

\(\displaystyle{ f(x)=lnln\frac{\sqrt{x^{2}+1} -x}{\sqrt{x^{2}-1} +x}}\)

\(\displaystyle{ f(x)=\frac{xsinx}{1+tgx}}\)

Obliczyć pochodne. Z góry dziękuje za waszą pomoc.

[M Ciesielski]

ad.2

\(\displaystyle{ f'(x) = \frac{(xcosx+sinx)(1+tgx) - xsinx\frac{1}{cos^2x}}{(1+tgx)^2}}\)

[miki999]

\(\displaystyle{ Zastosujemy\ podstawienia: \\ t=ln \frac{ \sqrt{x^{2}-1}-x }{ \sqrt{x^{2}-1}+x } \\ u= \frac{ \sqrt{x^{2}-1}-x }{ \sqrt{x^{2}-1}+x } \\ (ln(ln \frac{ \sqrt{x^{2}-1}-x }{ \sqrt{x^{2}-1}+x }))'= (lnt)' (ln \frac{ \sqrt{x^{2}-1}-x }{ \sqrt{x^{2}-1}+x })' = \frac{1}{t} (lnu)' ( \frac{ \sqrt{x^{2}-1}-x }{ \sqrt{x^{2}-1}+x } )' = \frac{1 }{ln \frac{ \sqrt{x^{2}-1}-x }{ \sqrt{x^{2}-1}+x }} \frac{1}{ \frac{ \sqrt{x^{2}-1}-x }{ \sqrt{x^{2}-1}+x }} \frac{( \sqrt{x^{2}+1}-x)' ( \sqrt{x^{2}-1}+x) - ( \sqrt{x^{2}+1}-x) ( \sqrt{x^{2}-1}+x)' }{ (\sqrt{x^{2}-1}+x)^{2} } = \frac{1 }{ln \frac{ \sqrt{x^{2}-1}-x }{ \sqrt{x^{2}-1}+x }} \frac{1}{ \frac{ \sqrt{x^{2}-1}-x }{ \sqrt{x^{2}-1}+x }} \frac{( \frac{2x}{ \sqrt{x^{2}-1} }-1) (\sqrt{x^{2}-1}+x)- (\frac{2x}{ \sqrt{x^{2}-1} }+1) ( \sqrt{x^{2}+1}-x) }{ (\sqrt{x^{2}-1}+x)^{2} }}\)

Pozdrawiam.