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

The derivative of (5e^xx^{3/2}/2+e^xx^{5/2}) is (20e^xx^{3/2}+15e^xx^{1/2}+4e^xx^{5/2})/4

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

The first derivative of (5e^xx^{3/2}/2+e^xx^{5/2}) is (20e^xx^{3/2}+15e^xx^{1/2}+4e^xx^{5/2})/4

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

\frac{20 e ^{x} x ^{\frac{3}{2}} + 15 e ^{x} x ^{\frac{1}{2}} + 4 e ^{x} x ^{\frac{5}{2}}}{4}

Solution steps

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

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

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

\frac{d}{d x} (\frac{5 e ^{x} x ^{\frac{3}{2}}}{2}) = \frac{5}{2} (e^{x} x^{\frac{3}{2}} + \frac{3 e ^{x} x ^{\frac{1}{2}}}{2})

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

= \frac{5}{2} (e^{x} x^{\frac{3}{2}} + \frac{3 e ^{x} x ^{\frac{1}{2}}}{2}) + e^{x} x^{\frac{5}{2}} + \frac{5 e ^{x} x ^{\frac{3}{2}}}{2}

Simplify

\frac{5}{2} (e^{x} x^{\frac{3}{2}} + \frac{3 e ^{x} x ^{\frac{1}{2}}}{2}) + e^{x} x^{\frac{5}{2}} + \frac{5 e ^{x} x ^{\frac{3}{2}}}{2} : \frac{20 e ^{x} x ^{\frac{3}{2}} + 15 e ^{x} x ^{\frac{1}{2}} + 4 e ^{x} x ^{\frac{5}{2}}}{4}

= \frac{20 e ^{x} x ^{\frac{3}{2}} + 15 e ^{x} x ^{\frac{1}{2}} + 4 e ^{x} x ^{\frac{5}{2}}}{4}

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What is the derivative of (5e^xx^{3/2}/2+e^xx^{5/2}) ?
The derivative of (5e^xx^{3/2}/2+e^xx^{5/2}) is (20e^xx^{3/2}+15e^xx^{1/2}+4e^xx^{5/2})/4
What is the first derivative of (5e^xx^{3/2}/2+e^xx^{5/2}) ?
The first derivative of (5e^xx^{3/2}/2+e^xx^{5/2}) is (20e^xx^{3/2}+15e^xx^{1/2}+4e^xx^{5/2})/4