# Question Differentiate the following functions: f(x) = ln ((x^5(3x-5)^2)/((x+1)^12(1-2x)^7)   g(x) = (e^x-ln(x))^tanx   (a) $$f(x)=\ln \left(\frac{x^{5}(3 x-5)^{2}}{(x+1)^{12}(1-2 x)^{7}}\right)$$ (b) $$g(x)=\left(e^{x}-\ln (x)\right)^{\tan x}$$.

Differentiate the following functions:

f(x) = ln ((x^5(3x-5)^2)/((x+1)^12(1-2x)^7)

g(x) = (e^x-ln(x))^tanx

Transcribed Image Text: (a) $$f(x)=\ln \left(\frac{x^{5}(3 x-5)^{2}}{(x+1)^{12}(1-2 x)^{7}}\right)$$ (b) $$g(x)=\left(e^{x}-\ln (x)\right)^{\tan x}$$.
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Transcribed Image Text: (a) $$f(x)=\ln \left(\frac{x^{5}(3 x-5)^{2}}{(x+1)^{12}(1-2 x)^{7}}\right)$$ (b) $$g(x)=\left(e^{x}-\ln (x)\right)^{\tan x}$$.
&#12304;General guidance&#12305;The answer provided below has been developed in a clear step by step manner.Step1/2)&#160;$$\mathrm{{f{{\left({x}\right)}}}={\ln{{\left(\frac{{{x}^{{5}}{\left({3}{x}-{5}\right)}^{{2}}}}{{{\left({x}+{1}\right)}^{{12}}{\left({1}-{2}{x}\right)}^{{7}}}}\right)}}}}$$Using property of logarthim $$\mathrm{{{\ln{{x}}}^{{n}}=}{n}{\ln{{x}}},{\ln{{\left(\frac{{m}}{{n}}\right)}}}={\ln{{m}}}-{\ln{{n}}}}$$$$\mathrm{{f{{\left({x}\right)}}}={5}{\ln{{x}}}+{2}{\ln{{\left({3}{x}-{5}\right)}}}-{12}{\ln{{\left({x}+{1}\right)}}}-{7}{\ln{{\left({1}-{2}{x}\right)}}}}$$By the Sum Rule, the derivative of $$\mathrm{{5}{\ln{{\left({x}\right)}}}+{2}{\ln{{\left({3}{x}-{5}\right)}}}-{12}{\ln{{\left({x}+{1}\right)}}}-{7}{\ln{{\left({1}-{2}{x}\right)}}}}$$ with respect to $$\mathrm{{x}}$$ is $$\mathrm{\frac{{{d}}}{{{\left.{d}{x}\right.}}}{\left[{5}{\ln{{\left({x}\right)}}}\right]}+\frac{{{d}}}{{{\left.{d}{x}\right.}}}{\left[{2}{\ln{{\left({3}{x}-{5}\right)}}}\right]}+\frac{{{d}}}{{{\left.{d}{x}\right.}}}{\left[-{12}{\ln{{\left({x}+{1}\right)}}}\right]}+\frac{{{d}}}{{{\left.{d}{x}\right.}}}{\left[-{7}{\ln{{\left({1}-{2}{x}\right)}}}\right]}}$$.$$\mathrm{\frac{{{d}}}{{{\left.{d}{x}\right.}}}{\left[{5}{\ln{{\left({x}\right)}}}\right]}+\frac{{{d}}}{{{\left.{d}{x}\right.}}}{\left[{2}{\ln{{\left({3}{x}-{5}\right)}}}\right]}+\frac{{{d}}}{{{\left.{d}{x}\right.}}}{\left[-{12}{\ln{{\left({x}+{1}\right)}}}\right]}+\frac{{{d}}}{{{\left.{d}{x}\right.}}}{\left[-{7}{\ln{{\left({1}-{2}{x}\right)}}}\right]}}$$Evaluate $$\mathrm{\frac{{{d}}}{{{\left.{d}{x}\right.}}}{\left[{5}{\ln{{\left({x}\right)}}}\right]}}$$.$$\mathrm{\frac{{{5}}}{{{x}}}+\frac{{{d}}}{{{\left.{d}{x}\right.}}}{\left[{2}{\ln{{\left({3}{x}-{5}\right)}}}\right]}+\frac{{{d}}}{{{\left.{d}{x}\right.}}}{\left[-{12}{\ln{{\left({x}+{1}\right)}}}\right]}+\frac{{{d}}}{{{\left.{d}{x}\right.}}}{\left[-{7}{\ln{{\left({1}-{2}{x}\right)}}}\right]}}$$Evaluate \( \mathr ... See the full answer