`y = log_5sqrt(x^2-1)` Find the derivative of the function

Friday, April 16, 2010

$\displaystyle{y}={\log}_{{5}}\sqrt{{{{x}}^{{2}}-{1}}}$ Find the derivative of the function

$\displaystyle{y}={\log}_{{5}}\sqrt{{{{x}}^{{2}}-{1}}}$

Before taking the derivative of the function, express the radical in exponent form.

$\displaystyle{y}={\log}_{{5}}{{\left({{x}}^{{2}}-{1}\right)}}^{{\frac{{1}}{{2}}}}$

Then, apply the logarithm rule $\displaystyle{\log}_{{b}}{\left({{a}}^{{m}}\right)}={m}\cdot{\log}_{{b}}{\left({a}\right)}$ .

$\displaystyle{y}=\frac{{1}}{{2}}{\log}_{{5}}{\left({{x}}^{{2}}-{1}\right)}$

From here, proceed to take the derivative of the function. Take note that the derivative formula of logarithm is $\displaystyle\frac{{d}}{{\left.{d}{x}}}{\left[{\log}_{{b}}{\left({u}\right)}\right]}=\frac{{1}}{{{\ln{{\left({b}\right)}}}\cdot{u}}}\cdot\frac{{{d}{u}}}{{\left.{d}{x}}}$ .

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

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{\left.{d}{x}}}=\frac{{d}}{{\left.{d}{x}}}{\left[\frac{{1}}{{2}}{\log}_{{5}}{\left({{x}}^{{2}}-{1}\right)}\right]}$

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{\left.{d}{x}}}=\frac{{1}}{{2}}\frac{{d}}{{\left.{d}{x}}}{\left[{\log}_{{5}}{\left({{x}}^{{2}}-{1}\right)}\right]}$

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{\left.{d}{x}}}=\frac{{1}}{{2}}\cdot\frac{{1}}{{{\ln{{\left({5}\right)}}}\cdot{\left({{x}}^{{2}}-{1}\right)}}}\cdot\frac{{d}}{{\left.{d}{x}}}{\left({{x}}^{{2}}-{1}\right)}$

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{\left.{d}{x}}}=\frac{{1}}{{2}}\cdot\frac{{1}}{{{\ln{{\left({5}\right)}}}\cdot{\left({{x}}^{{2}}-{1}\right)}}}\cdot{2}{x}$

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{\left.{d}{x}}}=\frac{{{2}{x}}}{{{2}{\left({{x}}^{{2}}-{1}\right)}{\ln{{\left({5}\right)}}}}}$

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{\left.{d}{x}}}=\frac{{x}}{{{\left({{x}}^{{2}}-{1}\right)}{\ln{{\left({5}\right)}}}}}$

Therefore, the derivative of the function is $\displaystyle\frac{{{\left.{d}{y}\right.}}}{{\left.{d}{x}}}=\frac{{x}}{{{\left({{x}}^{{2}}-{1}\right)}{\ln{{\left({5}\right)}}}}}$ .

Blog

Friday, April 16, 2010

$\displaystyle{y}={\log}_{{5}}\sqrt{{{{x}}^{{2}}-{1}}}$ Find the derivative of the function

No comments:

Post a Comment

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

Friday, April 16, 2010

Find the derivative of the function

No comments:

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?

$\displaystyle{y}={\log}_{{5}}\sqrt{{{{x}}^{{2}}-{1}}}$ 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?