Science:Math Exam Resources/Courses/MATH100/December 2011/Question 01 (j)/Solution 1

In this case we need to use logarithmic differentiation, indeed it is impossible to use the power rule since there is an $x^{2}$ as exponent. We cannot differentiate it in the same way we do for exponential functions since we take the power of x and not e or another constant. Hence we need to use logarithmic differentiation. First set

$y=x^{(x^{2})},$

then taking the logarithm on both sides we find

$\ln y=\ln(x^{(x^{2})})=x^{2}\ln(x).$

Now keeping in mind that y is a function of x, we can take the derivatives on both sides and get

${\frac {d}{dx}}\ln y={\frac {d}{dx}}(x^{2}\ln(x)).$

The left-hand side is simply

${\frac {d}{dx}}\ln y={\frac {1}{y}}\cdot {\frac {d}{dx}}y={\frac {1}{y}}y',$

while the right-hand side gives, using the product rule,

${\frac {d}{dx}}(x^{2}\ln(x))=2x\ln(x)+x^{2}{\frac {1}{x}}=2x\ln(x)+x.$

Putting everything back together, we get

${\frac {1}{y}}y'=2x\ln(x)+x,$

and thus

$\displaystyle y'=y(2x\ln(x)+x).$

Replacing y by $x^{(x^{2})},$ we finally get

$y'=x^{(x^{2})}(2x\ln(x)+x).$