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Popular >

derivative of (1+x^x)

Solution

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

Solution

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

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

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

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

Popular Examples

derivative of 1/2 bh derivative of

\frac{1}{2} bh

y^'+x^{-1}y=2xy^2

y^{^{'}} + x^{- 1} y = 2 x y^{2}

limit as x approaches 25/9 of 3(x^2)/2

x \to \frac{25}{9} lim (3 \frac{x ^{2}}{2})

integral of (tan(x))/(cos(x))

\int \frac{tan ( x )}{cos ( x )} d x

limit as x approaches 4 of 9*27

x \to 4 lim (9 \cdot 27)

Frequently Asked Questions (FAQ)

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