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

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

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

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

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derivative of

(x + 1)^{x + 1}

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

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

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

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

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