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Derivative of:
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Derivative of atan(x)
Identical expressions
log(x+ nine)^ five - five *x
logarithm of (x plus 9) to the power of 5 minus 5 multiply by x
logarithm of (x plus nine) to the power of five minus five multiply by x
log(x+9)5-5*x
logx+95-5*x
log(x+9)⁵-5*x
log(x+9)^5-5x
log(x+9)5-5x
logx+95-5x
logx+9^5-5x
Similar expressions
log(x+9)^5+5*x
log(x-9)^5-5*x

The derivative of the function/ log(x+9)^5-5*x

Derivative of log(x+9)^5-5*x

The solution

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   5             
log (x + 9) - 5*x

$$- 5 x + \log{\left(x + 9 \right)}^{5}$$

Transform

Detail solution

Differentiate $- 5 x + \log{\left(x + 9 \right)}^{5}$ term by term:
1. Let $u = \log{\left(x + 9 \right)}$ .
2. Apply the power rule: $u^{5}$ goes to $5 u^{4}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x + 9 \right)}$ :
  1. Let $u = x + 9$ .
  2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 9\right)$ :
    1. Differentiate $x + 9$ term by term:
      1. Apply the power rule: $x$ goes to $1$
      2. The derivative of the constant $9$ is zero.
      The result is: $1$
    The result of the chain rule is:
    
    $\frac{1}{x + 9}$
  The result of the chain rule is:
  
  $\frac{5 \log{\left(x + 9 \right)}^{4}}{x + 9}$
4. 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: $-5$
The result is: $-5 + \frac{5 \log{\left(x + 9 \right)}^{4}}{x + 9}$
Now simplify:

$\frac{5 \left(- x + \log{\left(x + 9 \right)}^{4} - 9\right)}{x + 9}$

The answer is:

$\frac{5 \left(- x + \log{\left(x + 9 \right)}^{4} - 9\right)}{x + 9}$

The graph

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The first derivative [src]

          4       
     5*log (x + 9)
-5 + -------------
         x + 9

$$-5 + \frac{5 \log{\left(x + 9 \right)}^{4}}{x + 9}$$

Simplify

The second derivative [src]

     3                        
5*log (9 + x)*(4 - log(9 + x))
------------------------------
                  2           
           (9 + x)

$$\frac{5 \left(4 - \log{\left(x + 9 \right)}\right) \log{\left(x + 9 \right)}^{3}}{\left(x + 9\right)^{2}}$$

Simplify

The third derivative [src]

      2        /       2                      \
10*log (9 + x)*\6 + log (9 + x) - 6*log(9 + x)/
-----------------------------------------------
                           3                   
                    (9 + x)

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

Simplify

The graph

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Derivative of log(x+9)^5-5*x

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The solution