Help Notes: `y = (cos(x))^x` Use logarithmic differentiation to find the derivative of the function.

Sunday, 30 January 2011

$\displaystyle{y}={{\left({\cos{{\left({x}\right)}}}\right)}}^{{x}}$ Use logarithmic differentiation to find the derivative of the function.

Differentiate
$\displaystyle{y}={{\left({\cos{{\left({x}\right)}}}\right)}}^{{x}}$ :

Take the natural logarithm of both sides:

$\displaystyle{\ln{{y}}}={{\ln{{\left({\cos{{\left({x}\right)}}}\right)}}}}^{{x}}$

Use the power property of logarithms:

$\displaystyle{\ln{{y}}}={x}{\ln{{\left({\cos{{\left({x}\right)}}}\right)}}}$

Differentiate; use the product rule on the
RHS:

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}{\left(\frac{{1}}{{y}}\right)}={\ln{{\left({\cos{{\left({x}\right)}}}\right)}}}+{x}\frac{{-{\sin{{\left({x}\right)}}}}}{{{\cos{{\left({x}\right)}}}}}$

$\displaystyle{y}'={y}{\left({\ln{{\left({\cos{{\left({x}\right)}}}\right)}}}-{x}{\tan{{\left({x}\right)}}}\right)}$

Substituting for y we get:

$\displaystyle{y}'={{\left({\cos{{\left({x}\right)}}}\right)}}^{{x}}{\left({\ln{{\left({\cos{{\left({x}\right)}}}\right)}}}-{x}{\tan{{\left({x}\right)}}}\right)}$

Help Notes

Sunday, 30 January 2011

$\displaystyle{y}={{\left({\cos{{\left({x}\right)}}}\right)}}^{{x}}$ Use logarithmic differentiation to find the derivative of the function.

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