What is the derivative of e^xx^e ?

The derivative of e^xx^e is e^xx^e+e^{1+x}x^{e-1}

What is the first derivative of e^xx^e ?

The first derivative of e^xx^e is e^xx^e+e^{1+x}x^{e-1}

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\frac{d}{d x} (e^{x} x^{e})

e^{x} x^{e} + e^{1 + x} x^{e - 1}

Solution steps

\frac{d}{d x} (e^{x} x^{e})

Apply the Product Rule:

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

f = e^{x}, g = x^{e}

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

\frac{d}{d x} (e^{x}) = e^{x}

\frac{d}{d x} (x^{e}) = e x^{e - 1}

= e^{x} x^{e} + e x^{e - 1} e^{x}

Apply exponent rule:

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

e e^{x} = e^{1 + x}

= e^{x} x^{e} + e^{1 + x} x^{e - 1}

\frac{d}{d x} (3^{s i n (x^{3})})

\int x^{5} \cdot e^{x} d x

\int \frac{t + 1}{t} d t

(x, y) \to (0, 0) lim ((x^{2} + y^{2}) ln (x^{2} + y^{2}))

y = 4 x - x^{2}

What is the derivative of e^xx^e ?
The derivative of e^xx^e is e^xx^e+e^{1+x}x^{e-1}
What is the first derivative of e^xx^e ?
The first derivative of e^xx^e is e^xx^e+e^{1+x}x^{e-1}