Derivative of ln(x+10)^11

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   11        
log  (x + 10)

$$\log{\left(x + 10 \right)}^{11}$$

log(x + 10)^11

Detail solution

Let $u = \log{\left(x + 10 \right)}$ .
Apply the power rule: $u^{11}$ goes to $11 u^{10}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x + 10 \right)}$ :
1. Let $u = x + 10$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 10\right)$ :
  1. Differentiate $x + 10$ term by term:
    1. Apply the power rule: $x$ goes to $1$
    2. The derivative of the constant $10$ is zero.
    The result is: $1$
  The result of the chain rule is:
  
  $\frac{1}{x + 10}$
The result of the chain rule is:

$\frac{11 \log{\left(x + 10 \right)}^{10}}{x + 10}$
Now simplify:

$\frac{11 \log{\left(x + 10 \right)}^{10}}{x + 10}$

The answer is:

$\frac{11 \log{\left(x + 10 \right)}^{10}}{x + 10}$

The first derivative [src]

      10        
11*log  (x + 10)
----------------
     x + 10

$$\frac{11 \log{\left(x + 10 \right)}^{10}}{x + 10}$$

Simplify

The second derivative [src]

      9                           
11*log (10 + x)*(10 - log(10 + x))
----------------------------------
                    2             
            (10 + x)

$$\frac{11 \left(10 - \log{\left(x + 10 \right)}\right) \log{\left(x + 10 \right)}^{9}}{\left(x + 10\right)^{2}}$$

Simplify

The third derivative [src]

      8         /        2                         \
22*log (10 + x)*\45 + log (10 + x) - 15*log(10 + x)/
----------------------------------------------------
                             3                      
                     (10 + x)

$$\frac{22 \left(\log{\left(x + 10 \right)}^{2} - 15 \log{\left(x + 10 \right)} + 45\right) \log{\left(x + 10 \right)}^{8}}{\left(x + 10\right)^{3}}$$

Simplify

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Derivative of ln(x+10)^11

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