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e^x*sqrt(x)
The derivative of the function/ e^x*sqrt(x)

Derivative of e^x*sqrt(x)

Function f() - derivative -N order at the point
v

The graph:

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The solution

You have entered [src] 
 x   ___
E *\/ x
exxe^{x} \sqrt{x} 
E^x*sqrt(x)
Detail solution

  1. Apply the product rule:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}

    f(x)=exf{\left(x \right)} = e^{x}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. The derivative of exe^{x} is itself.

    g(x)=xg{\left(x \right)} = \sqrt{x}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Apply the power rule: x\sqrt{x} goes to 12x\frac{1}{2 \sqrt{x}}

    The result is: xex+ex2x\sqrt{x} e^{x} + \frac{e^{x}}{2 \sqrt{x}}

  2. Now simplify:

    (x+12)exx\frac{\left(x + \frac{1}{2}\right) e^{x}}{\sqrt{x}}

The answer is:

(x+12)exx\frac{\left(x + \frac{1}{2}\right) e^{x}}{\sqrt{x}}

The graph
02468-8-6-4-2-10100100000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
               x  
  ___  x      e   
\/ x *e  + -------
               ___
           2*\/ x
xex+ex2x\sqrt{x} e^{x} + \frac{e^{x}}{2 \sqrt{x}}
Simplify
The second derivative [src] 
/  ___     1       1   \  x
|\/ x  + ----- - ------|*e 
|          ___      3/2|   
\        \/ x    4*x   /
(x+1x−14x32)ex\left(\sqrt{x} + \frac{1}{\sqrt{x}} - \frac{1}{4 x^{\frac{3}{2}}}\right) e^{x}
Simplify
The third derivative [src] 
/  ___     3         3        3   \  x
|\/ x  - ------ + ------- + ------|*e 
|           3/2       ___      5/2|   
\        4*x      2*\/ x    8*x   /
(x+32x−34x32+38x52)ex\left(\sqrt{x} + \frac{3}{2 \sqrt{x}} - \frac{3}{4 x^{\frac{3}{2}}} + \frac{3}{8 x^{\frac{5}{2}}}\right) e^{x}
Simplify
The graph
Derivative of e^x*sqrt(x)
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