Derivative of ln(x+4)^11

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   11       
log  (x + 4)

$$\log{\left(x + 4 \right)}^{11}$$

d /   11       \
--\log  (x + 4)/
dx

$$\frac{d}{d x} \log{\left(x + 4 \right)}^{11}$$

Transform

Detail solution

Let $u = \log{\left(x + 4 \right)}$ .
Apply the power rule: $u^{11}$ goes to $11 u^{10}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x + 4 \right)}$ :
1. Let $u = x + 4$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 4\right)$ :
  1. Differentiate $x + 4$ term by term:
    1. Apply the power rule: $x$ goes to $1$
    2. The derivative of the constant $4$ is zero.
    The result is: $1$
  The result of the chain rule is:
  
  $\frac{1}{x + 4}$
The result of the chain rule is:

$\frac{11 \log{\left(x + 4 \right)}^{10}}{x + 4}$
Now simplify:

$\frac{11 \log{\left(x + 4 \right)}^{10}}{x + 4}$

The answer is:

$\frac{11 \log{\left(x + 4 \right)}^{10}}{x + 4}$

The graph

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The first derivative [src]

      10       
11*log  (x + 4)
---------------
     x + 4

$$\frac{11 \log{\left(x + 4 \right)}^{10}}{x + 4}$$

Simplify

The second derivative [src]

      9                         
11*log (4 + x)*(10 - log(4 + x))
--------------------------------
                   2            
            (4 + x)

$$\frac{11 \cdot \left(10 - \log{\left(x + 4 \right)}\right) \log{\left(x + 4 \right)}^{9}}{\left(x + 4\right)^{2}}$$

Simplify

The third derivative [src]

      8        /        2                       \
22*log (4 + x)*\45 + log (4 + x) - 15*log(4 + x)/
-------------------------------------------------
                            3                    
                     (4 + x)

$$\frac{22 \left(\log{\left(x + 4 \right)}^{2} - 15 \log{\left(x + 4 \right)} + 45\right) \log{\left(x + 4 \right)}^{8}}{\left(x + 4\right)^{3}}$$

Simplify

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Derivative of ln(x+4)^11

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The solution