What is the derivative of a^xln(x) ?

The derivative of a^xln(x) is a^xln(a)ln(x)+(a^x)/x

What is the first derivative of a^xln(x) ?

The first derivative of a^xln(x) is a^xln(a)ln(x)+(a^x)/x

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\frac{d}{d x} (a^{x} ln (x))

a^{x} ln (a) ln (x) + \frac{a ^{x}}{x}

Solution steps

\frac{d}{d x} (a^{x} ln (x))

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = a^{x}, g = ln (x)

= \frac{d}{d x} (a^{x}) ln (x) + \frac{d}{d x} (ln (x)) a^{x}

\frac{d}{d x} (a^{x}) = a^{x} ln (a)

\frac{d}{d x} (ln (x)) = \frac{1}{x}

= a^{x} ln (a) ln (x) + \frac{1}{x} a^{x}

\frac{1}{x} a^{x} = \frac{a ^{x}}{x}

= a^{x} ln (a) ln (x) + \frac{a ^{x}}{x}

What is the derivative of a^xln(x) ?
The derivative of a^xln(x) is a^xln(a)ln(x)+(a^x)/x
What is the first derivative of a^xln(x) ?
The first derivative of a^xln(x) is a^xln(a)ln(x)+(a^x)/x