Derivative of y=log2(5xx+3)

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

$$\frac{\log{\left(x 5 x + 3 \right)}}{\log{\left(2 \right)}}$$

log((5*x)*x + 3)/log(2)

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = x 5 x + 3$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x 5 x + 3\right)$ :
  1. Differentiate $x 5 x + 3$ term by term:
    1. Apply the product rule:
      
      $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
      
      $f{\left(x \right)} = 5 x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $5$
      $g{\left(x \right)} = x$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
      1. Apply the power rule: $x$ goes to $1$
      The result is: $10 x$
    2. The derivative of the constant $3$ is zero.
    The result is: $10 x$
  The result of the chain rule is:
  
  $\frac{10 x}{x 5 x + 3}$
So, the result is: $\frac{10 x}{\left(x 5 x + 3\right) \log{\left(2 \right)}}$
Now simplify:

$\frac{10 x}{\left(5 x^{2} + 3\right) \log{\left(2 \right)}}$

The answer is:

$\frac{10 x}{\left(5 x^{2} + 3\right) \log{\left(2 \right)}}$

The graph

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Plot the graph f'(x)

The first derivative [src]

       10*x       
------------------
(5*x*x + 3)*log(2)

$$\frac{10 x}{\left(x 5 x + 3\right) \log{\left(2 \right)}}$$

Simplify

The second derivative [src]

    /          2  \
    |      10*x   |
-10*|-1 + --------|
    |            2|
    \     3 + 5*x /
-------------------
 /       2\        
 \3 + 5*x /*log(2)

$$- \frac{10 \left(\frac{10 x^{2}}{5 x^{2} + 3} - 1\right)}{\left(5 x^{2} + 3\right) \log{\left(2 \right)}}$$

Simplify

The third derivative [src]

      /          2  \
      |      20*x   |
100*x*|-3 + --------|
      |            2|
      \     3 + 5*x /
---------------------
            2        
  /       2\         
  \3 + 5*x / *log(2)

$$\frac{100 x \left(\frac{20 x^{2}}{5 x^{2} + 3} - 3\right)}{\left(5 x^{2} + 3\right)^{2} \log{\left(2 \right)}}$$

Simplify

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Derivative of y=log2(5xx+3)

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