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Popular >

derivative of x^pipi^x

Solution

derivative of

x^{π} π^{x}

Solution

π^{1 + x} x^{π - 1} + ln (π) π^{x} x^{π}

Solution steps

\frac{d}{d x} (x^{π} π^{x})

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{π}, g = π^{x}

= \frac{d}{d x} (x^{π}) π^{x} + \frac{d}{d x} (π^{x}) x^{π}

\frac{d}{d x} (x^{π}) = π x^{π - 1}

\frac{d}{d x} (π^{x}) = π^{x} ln (π)

= π x^{π - 1} π^{x} + π^{x} ln (π) x^{π}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

π π^{x} = π^{1 + x}

= π^{1 + x} x^{π - 1} + ln (π) π^{x} x^{π}

Popular Examples

derivative of f(x)=4x^{3/5}derivative of

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integral from-9 to 9 of sqrt(58)

\int_{- 9}^{9} 58

limit as x approaches 0 of (x+3)/x

x \to 0 lim (\frac{x + 3}{x})

limit as x approaches infinity of ln(0)

x \to \infty lim (ln (0))

integral of (sin^2(x)+cos^2(x))

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

Frequently Asked Questions (FAQ)

What is the derivative of x^pipi^x ?
The derivative of x^pipi^x is pi^{1+x}x^{pi-1}+ln(pi)pi^xx^pi
What is the first derivative of x^pipi^x ?
The first derivative of x^pipi^x is pi^{1+x}x^{pi-1}+ln(pi)pi^xx^pi