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

The derivative of x^nln(x) is nx^{n-1}ln(x)+x^{n-1}

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

The first derivative of x^nln(x) is nx^{n-1}ln(x)+x^{n-1}

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

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

Solution steps

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

Apply the Product Rule:

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

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

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

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

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

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

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

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

What is the derivative of x^nln(x) ?
The derivative of x^nln(x) is nx^{n-1}ln(x)+x^{n-1}
What is the first derivative of x^nln(x) ?
The first derivative of x^nln(x) is nx^{n-1}ln(x)+x^{n-1}