Find the differential coefficient of $a^{x} log x$ w.r.t $x$

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Solution

Let $y = a^{x} l o g x$
$∴$ on differentiation wrt $x$ , we have
$\frac{d y}{d x} = l o g x \times \frac{d (a^{x})}{d x} + a^{x} \times \frac{d (l o g x)}{d x}$ (using chain rule of differentiation)
$\frac{d y}{d x} = l o g x \times a^{x} (l n (a)) + a^{x} \times \frac{1}{x}$
$\frac{d y}{d x} = a^{x} (l o g x l n (a) + \frac{1}{x})$
This is the required coefficient.

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Find the differential coefficient of axlogx w.r.t x

Let y=axlogx∴ on differentiation wrt x, we havedydx=logx×d(ax)dx+ax×d(logx)dx (using chain rule of differentiation) dydx=logx×ax(ln(a))+ax×1xdydx=ax(logx ln(a)+1x)This is the required coefficient.

Find the differential coefficient of $a^{x} log x$ w.r.t $x$