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Derivative of y*lny
Derivative of 6x-ln(x+6)^6
Derivative of 3x^2
Derivative of pi*d^2/4
Identical expressions
six x-ln(x+ six)^6
6x minus ln(x plus 6) to the power of 6
six x minus ln(x plus six) to the power of 6
6x-ln(x+6)6
6x-lnx+66
6x-ln(x+6)⁶
6x-lnx+6^6
Similar expressions
6x-ln(x-6)^6
6x+ln(x+6)^6

The derivative of the function/ 6x-ln(x+6)^6

Derivative of 6x-ln(x+6)^6

The solution

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         6       
6*x - log (x + 6)

$$6 x - \log{\left(x + 6 \right)}^{6}$$

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Detail solution

Differentiate $6 x - \log{\left(x + 6 \right)}^{6}$ 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: $6$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \log{\left(x + 6 \right)}$ .
  2. Apply the power rule: $u^{6}$ goes to $6 u^{5}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x + 6 \right)}$ :
    1. Let $u = x + 6$ .
    2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 6\right)$ :
      1. Differentiate $x + 6$ term by term:
        
        Apply the power rule: $x$ goes to $1$
        
        The derivative of the constant $6$ is zero.
        
        The result is: $1$
      The result of the chain rule is:
      
      $\frac{1}{x + 6}$
    The result of the chain rule is:
    
    $\frac{6 \log{\left(x + 6 \right)}^{5}}{x + 6}$
  So, the result is: $- \frac{6 \log{\left(x + 6 \right)}^{5}}{x + 6}$
The result is: $6 - \frac{6 \log{\left(x + 6 \right)}^{5}}{x + 6}$
Now simplify:

$\frac{6 \left(x - \log{\left(x + 6 \right)}^{5} + 6\right)}{x + 6}$

The answer is:

$\frac{6 \left(x - \log{\left(x + 6 \right)}^{5} + 6\right)}{x + 6}$

The graph

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The first derivative [src]

         5       
    6*log (x + 6)
6 - -------------
        x + 6

$$6 - \frac{6 \log{\left(x + 6 \right)}^{5}}{x + 6}$$

Simplify

The second derivative [src]

     4                         
6*log (6 + x)*(-5 + log(6 + x))
-------------------------------
                   2           
            (6 + x)

$$\frac{6 \left(\log{\left(x + 6 \right)} - 5\right) \log{\left(x + 6 \right)}^{4}}{\left(x + 6\right)^{2}}$$

Simplify

The third derivative [src]

     3        /           2                       \
6*log (6 + x)*\-20 - 2*log (6 + x) + 15*log(6 + x)/
---------------------------------------------------
                             3                     
                      (6 + x)

$$\frac{6 \left(- 2 \log{\left(x + 6 \right)}^{2} + 15 \log{\left(x + 6 \right)} - 20\right) \log{\left(x + 6 \right)}^{3}}{\left(x + 6\right)^{3}}$$

Simplify

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Derivative of 6x-ln(x+6)^6

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