What is the derivative of x^aln(x) ?

The derivative of x^aln(x) is ax^{a-1}ln(x)+x^{a-1}

What is the first derivative of x^aln(x) ?

The first derivative of x^aln(x) is ax^{a-1}ln(x)+x^{a-1}

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

a x^{a - 1} ln (x) + x^{a - 1}

Solution steps

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

Apply the Product Rule:

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

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

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

\frac{d}{d x} (x^{a}) = a x^{a - 1}

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

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

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

= a x^{a - 1} ln (x) + x^{a - 1}

f (x) = 5 x e^{x}

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\frac{\partial}{\partial u ( v )} (\frac{e ^{v}}{u ( v ) + v ^{4}})

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\int csc^{4} (x) cot^{6} (x) d x

What is the derivative of x^aln(x) ?
The derivative of x^aln(x) is ax^{a-1}ln(x)+x^{a-1}
What is the first derivative of x^aln(x) ?
The first derivative of x^aln(x) is ax^{a-1}ln(x)+x^{a-1}