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Identical expressions
eleven *x-log(x+ four)^ eleven - three
11 multiply by x minus logarithm of (x plus 4) to the power of 11 minus 3
eleven multiply by x minus logarithm of (x plus four) to the power of eleven minus three
11*x-log(x+4)11-3
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11x-log(x+4)^11-3
11x-log(x+4)11-3
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Similar expressions
11*x-log(x+4)^11+3
11*x+log(x+4)^11-3
11*x-log(x-4)^11-3

The derivative of the function/ 11*x-log(x+4)^11-3

Derivative of 11*x-log(x+4)^11-3

The solution

You have entered [src]

          11           
11*x - log  (x + 4) - 3

$$\left(11 x - \log{\left(x + 4 \right)}^{11}\right) - 3$$

11*x - log(x + 4)^11 - 3

Detail solution

Differentiate $\left(11 x - \log{\left(x + 4 \right)}^{11}\right) - 3$ term by term:
1. Differentiate $11 x - \log{\left(x + 4 \right)}^{11}$ 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: $x$ goes to $1$
    So, the result is: $11$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = \log{\left(x + 4 \right)}$ .
    2. Apply the power rule: $u^{11}$ goes to $11 u^{10}$
    3. 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 $\frac{d}{d x} \left(x + 4\right)$ :
        
        Differentiate $x + 4$ term by term:
        
        Apply the power rule: $x$ goes to $1$
        
        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:
      
      $\frac{11 \log{\left(x + 4 \right)}^{10}}{x + 4}$
    So, the result is: $- \frac{11 \log{\left(x + 4 \right)}^{10}}{x + 4}$
  The result is: $11 - \frac{11 \log{\left(x + 4 \right)}^{10}}{x + 4}$
2. The derivative of the constant $-3$ is zero.
The result is: $11 - \frac{11 \log{\left(x + 4 \right)}^{10}}{x + 4}$
Now simplify:

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

The answer is:

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

The graph

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The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

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

The graph

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Derivative of 11*x-log(x+4)^11-3

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The solution