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

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  _________________
\/ (x - 3)*(x + 3)

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

sqrt((x - 3)*(x + 3))

Detail solution

Let $u = \left(x - 3\right) \left(x + 3\right)$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 3\right) \left(x + 3\right)$ :
1. 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: $1$
  $g{\left(x \right)} = x + 3$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  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: $1$
  The result is: $2 x$
The result of the chain rule is:

$\frac{x}{\sqrt{\left(x - 3\right) \left(x + 3\right)}}$
Now simplify:

$\frac{x}{\sqrt{x^{2} - 9}}$

The answer is:

$\frac{x}{\sqrt{x^{2} - 9}}$

The graph

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The first derivative [src]

    _________________
x*\/ (x - 3)*(x + 3) 
---------------------
   (x - 3)*(x + 3)

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

Simplify

The second derivative [src]

                     /                             2       \
  __________________ |      x        x            x        |
\/ (-3 + x)*(3 + x) *|1 - ------ - ----- + ----------------|
                     \    -3 + x   3 + x   (-3 + x)*(3 + x)/
------------------------------------------------------------
                      (-3 + x)*(3 + x)

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

Simplify

The third derivative [src]

                     /                                                   3                     2                   2                         \
  __________________ |    2        2        2*x        2*x              x                   3*x                 3*x                5*x       |
\/ (-3 + x)*(3 + x) *|- ------ - ----- + --------- + -------- + ------------------ - ----------------- - ----------------- + ----------------|
                     |  -3 + x   3 + x           2          2           2        2                   2           2           (-3 + x)*(3 + x)|
                     \                   (-3 + x)    (3 + x)    (-3 + x) *(3 + x)    (-3 + x)*(3 + x)    (-3 + x) *(3 + x)                   /
----------------------------------------------------------------------------------------------------------------------------------------------
                                                               (-3 + x)*(3 + x)

$$\frac{\sqrt{\left(x - 3\right) \left(x + 3\right)} \left(\frac{x^{3}}{\left(x - 3\right)^{2} \left(x + 3\right)^{2}} - \frac{3 x^{2}}{\left(x - 3\right) \left(x + 3\right)^{2}} - \frac{3 x^{2}}{\left(x - 3\right)^{2} \left(x + 3\right)} + \frac{2 x}{\left(x + 3\right)^{2}} + \frac{5 x}{\left(x - 3\right) \left(x + 3\right)} + \frac{2 x}{\left(x - 3\right)^{2}} - \frac{2}{x + 3} - \frac{2}{x - 3}\right)}{\left(x - 3\right) \left(x + 3\right)}$$

Simplify

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

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