Mister Exam

Other calculators

3sqrtx*e^(-x)
The derivative of the function/ 3sqrtx*e^(-x)

Derivative of 3sqrtx*e^(-x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
    ___  -x
3*\/ x *e
3xe−x3 \sqrt{x} e^{- x} 
d /    ___  -x\
--\3*\/ x *e  /
dx
ddx3xe−x\frac{d}{d x} 3 \sqrt{x} e^{- x}
Transform
Detail solution

  1. The derivative of a constant times a function is the constant times the derivative of the function.

    1. Apply the quotient rule, which is:

      ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}

      f(x)=xf{\left(x \right)} = \sqrt{x} and g(x)=exg{\left(x \right)} = e^{x}.

      To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

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

      To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

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

      Now plug in to the quotient rule:

      (−xex+ex2x)e−2x\left(- \sqrt{x} e^{x} + \frac{e^{x}}{2 \sqrt{x}}\right) e^{- 2 x}

    So, the result is: 3(−xex+ex2x)e−2x3 \left(- \sqrt{x} e^{x} + \frac{e^{x}}{2 \sqrt{x}}\right) e^{- 2 x}

  2. Now simplify:

    3⋅(1−2x)e−x2x\frac{3 \cdot \left(1 - 2 x\right) e^{- x}}{2 \sqrt{x}}

The answer is:

3⋅(1−2x)e−x2x\frac{3 \cdot \left(1 - 2 x\right) e^{- x}}{2 \sqrt{x}}

The graph
02468-8-6-4-2-10105-5
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
                    -x 
      ___  -x    3*e   
- 3*\/ x *e   + -------
                    ___
                2*\/ x
−3xe−x+3e−x2x- 3 \sqrt{x} e^{- x} + \frac{3 e^{- x}}{2 \sqrt{x}}
Simplify
The second derivative [src] 
  /  ___     1       1   \  -x
3*|\/ x  - ----- - ------|*e  
  |          ___      3/2|    
  \        \/ x    4*x   /
3(x−1x−14x32)e−x3 \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
3*|- \/ x  + ------- + ------ + ------|*e  
  |              ___      3/2      5/2|    
  \          2*\/ x    4*x      8*x   /
3(−x+32x+34x32+38x52)e−x3 \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 3sqrtx*e^(-x)
    © Mister Exam – Calculator