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Popular >

derivative of y=(x+1)^x

Solution

derivative of

y = (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 from 0 to 2 of 2pix(2x^2-x^3)

\int_{0}^{2} 2 π x (2 x^{2} - x^{3}) d x

integral of 1/((a+b)-(a-b)x^2)

\int \frac{1}{( a + b ) - ( a - b ) x ^{2}} d x

integral of (22)/((1-x^2)^{3/2)}

\int \frac{22}{( 1 - x ^{2} ) ^{\frac{3}{2}}} d x

(\partial)/(\partial x)(2xarcsin(x/y))

\frac{\partial}{\partial x} (2 x arcsin (\frac{x}{y}))

area y=x^{-3},1<= x<= 5 area

y = x^{- 3}, 1 \leq x \leq 5

Frequently Asked Questions (FAQ)

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