Question

If y = a^(a^x) , then dy/dx =

Answer 1

$take \: \\ y = {a}^{ ({a}^{x}) } \\ taking \: logarithm \: both \: sides \\ \\ ln(y) = ln( {a}^{ ({a}^{x}) }) \\ using \: property \: of \: log - \\ ln( {a}^{x} ) = x ln(a) \\ \\ ln(y) = {a}^{x} ln(a) \\ \\ again \: taking \: logaritm \\ ln( ln(y) ) = ln( {a}^{x} ln(a) ) \\ ln( ln(y) ) = ln( {a}^{x} ) + ln( ln(a) ) \\ ln( ln(y) ) = x ln(a) + ln( ln(a) ) \\ on \: diffrentiating \: both \: sides \: wrt \: x \\ \frac{1}{ ln(y) } . \frac{1}{y} . \frac{dy}{dx} = ln(a) + 0 \\ \frac{dy}{dx} = y. ln(a) . ln(y ) \\ putting \: vaue \: of \: y = {a}^{ ({a}^{x}) } \: \\ \\ \frac{dy}{dx} = {a}^{ ({a}^{x}) }. ln(a) . ln( {a}^{ ({a}^{x}) }) \:$



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