`y = ln(ln(x^2))` Find the derivative of the function.

Friday, March 19, 2010

$\displaystyle{y}={\ln{{\left({\ln{{\left({{x}}^{{2}}\right)}}}\right)}}}$

To take the derivative of this, use the formula:

$\displaystyle{\left({\ln{{u}}}\right)}'=\frac{{1}}{{u}}\cdot{u}'$

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

$\displaystyle{y}'=\frac{{1}}{{{\ln{{\left({{x}}^{{2}}\right)}}}}}\cdot{\left({\ln{{\left({{x}}^{{2}}\right)}}}\right)}'$

$\displaystyle{y}'=\frac{{1}}{{{\ln{{\left({{x}}^{{2}}\right)}}}}}\cdot\frac{{1}}{{{x}}^{{2}}}\cdot{\left({{x}}^{{2}}\right)}'$

To take the derivative of the innermost function, apply the formula $\displaystyle{\left({{x}}^{{n}}\right)}'={n}\cdot{{x}}^{{{n}-{1}}}$ .

$\displaystyle{y}'=\frac{{1}}{{{\ln{{\left({{x}}^{{2}}\right)}}}}}\cdot\frac{{1}}{{{x}}^{{2}}}\cdot{2}{x}$

$\displaystyle{y}'=\frac{{2}}{{{x}{\ln{{\left({{x}}^{{2}}\right)}}}}}$

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

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