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Identical expressions
log10(x+ seven)^ nine -9x
logarithm of 10(x plus 7) to the power of 9 minus 9x
logarithm of 10(x plus seven) to the power of nine minus 9x
log10(x+7)9-9x
log10x+79-9x
log10(x+7)⁹-9x
log10x+7^9-9x
Similar expressions
log10(x+7)^9+9x
log10(x-7)^9-9x

The derivative of the function/ log10(x+7)^9-9x

Derivative of log10(x+7)^9-9x

The solution

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            9      
/log(x + 7)\       
|----------|  - 9*x
\ log(10)  /

$$\left(\frac{\log{\left(x + 7 \right)}}{\log{\left(10 \right)}}\right)^{9} - 9 x$$

  /            9      \
d |/log(x + 7)\       |
--||----------|  - 9*x|
dx\\ log(10)  /       /

$$\frac{d}{d x} \left(\left(\frac{\log{\left(x + 7 \right)}}{\log{\left(10 \right)}}\right)^{9} - 9 x\right)$$

Transform

Detail solution

Differentiate $- 9 x + \left(\frac{\log{\left(x + 7 \right)}}{\log{\left(10 \right)}}\right)^{9}$ term by term:
1. Let $u = \frac{\log{\left(x + 7 \right)}}{\log{\left(10 \right)}}$ .
2. Apply the power rule: $u^{9}$ goes to $9 u^{8}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{\log{\left(x + 7 \right)}}{\log{\left(10 \right)}}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = x + 7$ .
    2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 7\right)$ :
      1. Differentiate $x + 7$ term by term:
        
        Apply the power rule: $x$ goes to $1$
        
        The derivative of the constant $7$ is zero.
        
        The result is: $1$
      The result of the chain rule is:
      
      $\frac{1}{x + 7}$
    So, the result is: $\frac{1}{\left(x + 7\right) \log{\left(10 \right)}}$
  The result of the chain rule is:
  
  $\frac{9 \log{\left(x + 7 \right)}^{8}}{\left(x + 7\right) \log{\left(10 \right)}^{9}}$
4. The derivative of a constant times a function is the constant times the derivative of the function.
  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: $9$
  So, the result is: $-9$
The result is: $-9 + \frac{9 \log{\left(x + 7 \right)}^{8}}{\left(x + 7\right) \log{\left(10 \right)}^{9}}$
Now simplify:

$\frac{9 \left(- \left(x + 7\right) \log{\left(10 \right)}^{9} + \log{\left(x + 7 \right)}^{8}\right)}{\left(x + 7\right) \log{\left(10 \right)}^{9}}$

The answer is:

$\frac{9 \left(- \left(x + 7\right) \log{\left(10 \right)}^{9} + \log{\left(x + 7 \right)}^{8}\right)}{\left(x + 7\right) \log{\left(10 \right)}^{9}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

            9          
         log (x + 7)   
       9*-----------   
              9        
           log (10)    
-9 + ------------------
     (x + 7)*log(x + 7)

$$-9 + \frac{9 \frac{\log{\left(x + 7 \right)}^{9}}{\log{\left(10 \right)}^{9}}}{\left(x + 7\right) \log{\left(x + 7 \right)}}$$

Simplify

The second derivative [src]

     7                        
9*log (7 + x)*(8 - log(7 + x))
------------------------------
             2    9           
      (7 + x) *log (10)

$$\frac{9 \cdot \left(- \log{\left(x + 7 \right)} + 8\right) \log{\left(x + 7 \right)}^{7}}{\left(x + 7\right)^{2} \log{\left(10 \right)}^{9}}$$

Simplify

The third derivative [src]

      6        /        2                       \
18*log (7 + x)*\28 + log (7 + x) - 12*log(7 + x)/
-------------------------------------------------
                       3    9                    
                (7 + x) *log (10)

$$\frac{18 \left(\log{\left(x + 7 \right)}^{2} - 12 \log{\left(x + 7 \right)} + 28\right) \log{\left(x + 7 \right)}^{6}}{\left(x + 7\right)^{3} \log{\left(10 \right)}^{9}}$$

Simplify

The graph

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Derivative of log10(x+7)^9-9x

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The solution