What is the derivative of (x^3e^x/(sqrt(x))) ?

The derivative of (x^3e^x/(sqrt(x))) is e^xx^{5/2}+(5e^xx^{3/2})/2

What is the first derivative of (x^3e^x/(sqrt(x))) ?

The first derivative of (x^3e^x/(sqrt(x))) is e^xx^{5/2}+(5e^xx^{3/2})/2

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\frac{d}{d x} (\frac{x ^{3} e ^{x}}{x})

e^{x} x^{\frac{5}{2}} + \frac{5 e ^{x} x ^{\frac{3}{2}}}{2}

Solution steps

\frac{d}{d x} (\frac{x ^{3} e ^{x}}{x})

Simplify

\frac{x ^{3} e ^{x}}{x} : e^{x} x^{\frac{5}{2}}

= \frac{d}{d x} (e^{x} x^{\frac{5}{2}})

Apply the Product Rule:

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

f = e^{x}, g = x^{\frac{5}{2}}

= \frac{d}{d x} (e^{x}) x^{\frac{5}{2}} + \frac{d}{d x} (x^{\frac{5}{2}}) e^{x}

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

\frac{d}{d x} (x^{\frac{5}{2}}) = \frac{5 x ^{\frac{3}{2}}}{2}

= e^{x} x^{\frac{5}{2}} + \frac{5 x ^{\frac{3}{2}}}{2} e^{x}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= e^{x} x^{\frac{5}{2}} + \frac{5 e ^{x} x ^{\frac{3}{2}}}{2}

\frac{d}{d y} (\frac{x ln ( \frac{x y}{z} )}{z})

d er i v a t i v e 12 x - 2 x

\int - \frac{1}{k} cos (k x) d x

\frac{\partial}{\partial y} (6 y e^{2 x y} + x)

x \to 0 - lim (\frac{x}{∣ x ∣})

What is the derivative of (x^3e^x/(sqrt(x))) ?
The derivative of (x^3e^x/(sqrt(x))) is e^xx^{5/2}+(5e^xx^{3/2})/2
What is the first derivative of (x^3e^x/(sqrt(x))) ?
The first derivative of (x^3e^x/(sqrt(x))) is e^xx^{5/2}+(5e^xx^{3/2})/2