Derivative of ln(x+5)^9

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   9       
log (x + 5)

$$\log{\left(x + 5 \right)}^{9}$$

log(x + 5)^9

Detail solution

Let $u = \log{\left(x + 5 \right)}$ .
Apply the power rule: $u^{9}$ goes to $9 u^{8}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x + 5 \right)}$ :
1. Let $u = x + 5$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 5\right)$ :
  1. Differentiate $x + 5$ term by term:
    1. Apply the power rule: $x$ goes to $1$
    2. The derivative of the constant $5$ is zero.
    The result is: $1$
  The result of the chain rule is:
  
  $\frac{1}{x + 5}$
The result of the chain rule is:

$\frac{9 \log{\left(x + 5 \right)}^{8}}{x + 5}$
Now simplify:

$\frac{9 \log{\left(x + 5 \right)}^{8}}{x + 5}$

The answer is:

$\frac{9 \log{\left(x + 5 \right)}^{8}}{x + 5}$

The graph

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The first derivative [src]

     8       
9*log (x + 5)
-------------
    x + 5

$$\frac{9 \log{\left(x + 5 \right)}^{8}}{x + 5}$$

Simplify

The second derivative [src]

     7                        
9*log (5 + x)*(8 - log(5 + x))
------------------------------
                  2           
           (5 + x)

$$\frac{9 \left(8 - \log{\left(x + 5 \right)}\right) \log{\left(x + 5 \right)}^{7}}{\left(x + 5\right)^{2}}$$

Simplify

The third derivative [src]

      6        /        2                       \
18*log (5 + x)*\28 + log (5 + x) - 12*log(5 + x)/
-------------------------------------------------
                            3                    
                     (5 + x)

$$\frac{18 \left(\log{\left(x + 5 \right)}^{2} - 12 \log{\left(x + 5 \right)} + 28\right) \log{\left(x + 5 \right)}^{6}}{\left(x + 5\right)^{3}}$$

Simplify

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Derivative of ln(x+5)^9

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The solution