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

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

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

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

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

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

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

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

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

\frac{d y}{d x} + (3 x y^{3}) = 0, y (0) = 1

\frac{d}{d x} (\frac{3}{6 x + 3})

\int_{0}^{5} \frac{1}{x ^{\frac{21}{22}}} d x

\int \frac{sin ( 2 x )}{42 + cos ^{2} ( x )} d x

\int \frac{x ^{3} - 4 x + 1}{x ^{2} - 3 x + 2} d x

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