What is the derivative of (x^{x+1}) ?

The derivative of (x^{x+1}) is x^{x+1}(ln(x)+(x+1)/x)

What is the first derivative of (x^{x+1}) ?

The first derivative of (x^{x+1}) is x^{x+1}(ln(x)+(x+1)/x)

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\frac{d}{d x} ((x)^{x + 1})

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

Solution steps

\frac{d}{d x} ((x)^{x + 1})

Apply exponent rule:

a^{b} = e^{b l n (a)}

x^{x + 1} = e^{(x + 1) l n (x)}

= \frac{d}{d x} (e^{(x + 1) l n (x)})

Apply the chain rule:

e^{(x + 1) l n (x)} \frac{d}{d x} ((x + 1) ln (x))

= e^{(x + 1) l n (x)} \frac{d}{d x} ((x + 1) ln (x))

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

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

e^{(x + 1) l n (x)} = x^{x + 1}

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

What is the derivative of (x^{x+1}) ?
The derivative of (x^{x+1}) is x^{x+1}(ln(x)+(x+1)/x)
What is the first derivative of (x^{x+1}) ?
The first derivative of (x^{x+1}) is x^{x+1}(ln(x)+(x+1)/x)