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Derivative of:
Derivative of sin(x)^2
Derivative of e^(2-x)
Derivative of cos(x/3)
Derivative of √x
Identical expressions
ln(-x^ two + nine)
ln( minus x squared plus 9)
ln( minus x to the power of two plus nine)
ln(-x2+9)
ln-x2+9
ln(-x²+9)
ln(-x to the power of 2+9)
ln-x^2+9
Similar expressions
ln(x^2+9)
ln(-x^2-9)

The derivative of the function/ ln(-x^2+9)

Derivative of ln(-x^2+9)

The solution

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   /   2    \
log\- x  + 9/

$$\log{\left(9 - x^{2} \right)}$$

log(-x^2 + 9)

Detail solution

Let $u = 9 - x^{2}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(9 - x^{2}\right)$ :
1. Differentiate $9 - x^{2}$ term by term:
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    So, the result is: $- 2 x$
  2. The derivative of the constant $9$ is zero.
  The result is: $- 2 x$
The result of the chain rule is:

$- \frac{2 x}{9 - x^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  -2*x  
--------
   2    
- x  + 9

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

Simplify

The second derivative [src]

  /         2 \
  |      2*x  |
2*|1 - -------|
  |          2|
  \    -9 + x /
---------------
          2    
    -9 + x

$$\frac{2 \left(- \frac{2 x^{2}}{x^{2} - 9} + 1\right)}{x^{2} - 9}$$

Simplify

The third derivative [src]

    /          2 \
    |       4*x  |
4*x*|-3 + -------|
    |           2|
    \     -9 + x /
------------------
             2    
    /      2\     
    \-9 + x /

$$\frac{4 x \left(\frac{4 x^{2}}{x^{2} - 9} - 3\right)}{\left(x^{2} - 9\right)^{2}}$$

Simplify

The graph

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Derivative of ln(-x^2+9)

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The solution