Mister Exam

Derivative of (x-3)*sqrt(x)

Function f() - derivative -N order at the point
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The graph:

from to

Piecewise:

The solution

You have entered [src]
          ___
(x - 3)*\/ x 
x(x3)\sqrt{x} \left(x - 3\right)
(x - 3)*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)=x3f{\left(x \right)} = x - 3; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Differentiate x3x - 3 term by term:

      1. Apply the power rule: xx goes to 11

      2. The derivative of the constant 3-3 is zero.

      The result is: 11

    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: x+x32x\sqrt{x} + \frac{x - 3}{2 \sqrt{x}}

  2. Now simplify:

    3(x1)2x\frac{3 \left(x - 1\right)}{2 \sqrt{x}}


The answer is:

3(x1)2x\frac{3 \left(x - 1\right)}{2 \sqrt{x}}

The graph
02468-8-6-4-2-1010-2525
The first derivative [src]
  ___    x - 3 
\/ x  + -------
            ___
        2*\/ x 
x+x32x\sqrt{x} + \frac{x - 3}{2 \sqrt{x}}
The second derivative [src]
    -3 + x
1 - ------
     4*x  
----------
    ___   
  \/ x    
1x34xx\frac{1 - \frac{x - 3}{4 x}}{\sqrt{x}}
The third derivative [src]
  /     -3 + x\
3*|-2 + ------|
  \       x   /
---------------
        3/2    
     8*x       
3(2+x3x)8x32\frac{3 \left(-2 + \frac{x - 3}{x}\right)}{8 x^{\frac{3}{2}}}
The graph
Derivative of (x-3)*sqrt(x)