ln Diff Example

[Jason76]

\(\displaystyle y = (\cos(4x))^{x}\)

\(\displaystyle \ln[y] = \ln [(\cos(4x))^{x}]\)

\(\displaystyle \ln[y] = x \ln[(\cos(4x))]\)

\(\displaystyle \dfrac{1}{y}y' = [\ln(\cos (4x))][1] + [x][\dfrac{1}{\cos(4x)} \)

\(\displaystyle \dfrac{1}{y}y' = [\ln(\cos (4x))][1] + [x][\dfrac{1}{[\cos(u) du]} \)

\(\displaystyle \dfrac{1}{y}y' = [\ln(\cos (4x))][1] + [x][\dfrac{1}{[\cos(u) (4)]} \)

\(\displaystyle \dfrac{1}{y}y' = [\ln(\cos (4x))][1] + [x][\dfrac{1}{4 \cos(4x)}] \)

\(\displaystyle y' = [y] \ln(\cos (4x))+ \dfrac{x}{4 \cos(4x)} \)

\(\displaystyle y' = [ (\cos(4x))^{x}] \ln(\cos (4x))+ x \dfrac{1}{4 \cos(4x)} \)

 

[Deleted member 4993]

\(\displaystyle y = (\cos(4x))^{x}\)

\(\displaystyle \ln[y] = \ln [(\cos(4x))^{x}]\)

\(\displaystyle \ln[y] = x \ln[(\cos(4x))]\)

\(\displaystyle \dfrac{1}{y}y' = [\ln(\cos (4x))][1] + [x][\dfrac{1}{\cos(4x)} \) ............ Incorrect .............. should be

\(\displaystyle \dfrac{1}{y}y' = [\ln(\cos (4x))][1] + [x][\dfrac{1}{\cos(4x)}] * [-sin(4x)] * [4] \)

\(\displaystyle \dfrac{1}{y}y' = [\ln(\cos (4x))][1] + [x][\dfrac{1}{[\cos(u) du]} \)

\(\displaystyle \dfrac{1}{y}y' = [\ln(\cos (4x))][1] + [x][\dfrac{1}{[\cos(u) (4)]} \)

\(\displaystyle \dfrac{1}{y}y' = [\ln(\cos (4x))][1] + [x][\dfrac{1}{4 \cos(4x)}] \)

\(\displaystyle y' = [y] \ln(\cos (4x))+ \dfrac{x}{4 \cos(4x)} \)

\(\displaystyle y' = [ (\cos(4x))^{x}] \ln(\cos (4x))+ x \dfrac{1}{4 \cos(4x)} \)

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