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Popular >

derivative of (x+1^x)

Solution

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

Solution

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

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

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

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

Popular Examples

integral of ((y^4+2y^2-1))/(sqrt(y))

\int \frac{( y ^{4} + 2 y ^{2} - 1 )}{y} d y

integral of-1/(1+e^x)

\int - \frac{1}{1 + e ^{x}} d x

integral of e^{100t}*60

\int e^{100 t} \cdot 60 d t

limit as x approaches 3 of 4/(x-3)

x \to 3 lim (\frac{4}{x - 3})

derivative of f(x)=3(x-2)^2 derivative of

f (x) = 3 (x - 2)^{2}

Frequently Asked Questions (FAQ)

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