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The derivative of the function/ (x-3)*sqrt(x)

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

The solution

You have entered [src]

          ___
(x - 3)*\/ x

$$\sqrt{x} \left(x - 3\right)$$

(x - 3)*sqrt(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x - 3$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x - 3$ term by term:
  1. Apply the power rule: $x$ goes to $1$
  2. The derivative of the constant $-3$ is zero.
  The result is: $1$
$g{\left(x \right)} = \sqrt{x}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
The result is: $\sqrt{x} + \frac{x - 3}{2 \sqrt{x}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  ___    x - 3 
\/ x  + -------
            ___
        2*\/ x

$$\sqrt{x} + \frac{x - 3}{2 \sqrt{x}}$$

Simplify

The second derivative [src]

    -3 + x
1 - ------
     4*x  
----------
    ___   
  \/ x

$$\frac{1 - \frac{x - 3}{4 x}}{\sqrt{x}}$$

Simplify

The third derivative [src]

  /     -3 + x\
3*|-2 + ------|
  \       x   /
---------------
        3/2    
     8*x

$$\frac{3 \left(-2 + \frac{x - 3}{x}\right)}{8 x^{\frac{3}{2}}}$$

Simplify

The graph

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