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

The derivative of x^{sqrt(x)+1} is (x^{sqrt(x)}(sqrt(x)ln(x)+2(1+sqrt(x))))/2

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

The first derivative of x^{sqrt(x)+1} is (x^{sqrt(x)}(sqrt(x)ln(x)+2(1+sqrt(x))))/2

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

\frac{x ^{x} ( x ln ( x ) + 2 ( 1 + x ) )}{2}

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)) = \frac{x ln ( x ) + 2 ( 1 + x )}{2 x}

= e^{(x + 1) l n (x)} \frac{x ln ( x ) + 2 ( 1 + x )}{2 x}

Simplify

e^{(x + 1) l n (x)} \frac{x ln ( x ) + 2 ( 1 + x )}{2 x} : \frac{x ^{x} ( x ln ( x ) + 2 ( 1 + x ) )}{2}

= \frac{x ^{x} ( x ln ( x ) + 2 ( 1 + x ) )}{2}

\frac{d}{d x} (- e^{- x y})

\int_{1}^{\infty} (e^{- x}) d x

\int \frac{6}{x - 1} d x

\int 3 x + 2 d x

What is the derivative of x^{sqrt(x)+1} ?
The derivative of x^{sqrt(x)+1} is (x^{sqrt(x)}(sqrt(x)ln(x)+2(1+sqrt(x))))/2
What is the first derivative of x^{sqrt(x)+1} ?
The first derivative of x^{sqrt(x)+1} is (x^{sqrt(x)}(sqrt(x)ln(x)+2(1+sqrt(x))))/2