`y = log_4(5x+1)` Find the derivative of the function

Tuesday, May 12, 2009

$\displaystyle{y}={\log}_{{4}}{\left({5}{x}+{1}\right)}$ Find the derivative of the function

$\displaystyle{y}={\log}_{{4}}{\left({5}{x}+{1}\right)}$

The derivative formula of a logarithm is:

$\displaystyle\frac{{d}}{{{\left.{d}{x}\right.}}}{\left[{\log}_{{a}}{\left({u}\right)}\right]}=\frac{{1}}{{{\ln{{\left({a}\right)}}}\cdot{u}}}\cdot\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}$

Applying that formula, the derivative of the function will be:

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}=\frac{{d}}{{{\left.{d}{x}\right.}}}{\left[{\log}_{{4}}{\left({5}{x}+{1}\right)}\right]}$

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}=\frac{{1}}{{{\ln{{\left({4}\right)}}}\cdot{\left({5}{x}+{1}\right)}}}\cdot\frac{{d}}{{{\left.{d}{x}\right.}}}{\left({5}{x}+{1}\right)}$

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}=\frac{{1}}{{{\ln{{\left({4}\right)}}}\cdot{\left({5}{x}+{1}\right)}}}\cdot{5}$

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}=\frac{{5}}{{{\left({5}{x}+{1}\right)}{\ln{{\left({4}\right)}}}}}$

Therefore, the derivative of the function is $\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}=\frac{{5}}{{{\left({5}{x}+{1}\right)}{\ln{{\left({4}\right)}}}}}$ .

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Tuesday, May 12, 2009

$\displaystyle{y}={\log}_{{4}}{\left({5}{x}+{1}\right)}$ Find the derivative of the function

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Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

Tuesday, May 12, 2009

Find the derivative of the function

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Post a Comment

Thomas Jefferson&#39;s election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

$\displaystyle{y}={\log}_{{4}}{\left({5}{x}+{1}\right)}$ Find the derivative of the function

Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?