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(two *x+ three)/(log(x)/log(three))
(2 multiply by x plus 3) divide by ( logarithm of (x) divide by logarithm of (3))
(two multiply by x plus three) divide by ( logarithm of (x) divide by logarithm of (three))
(2x+3)/(log(x)/log(3))
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(2*x+3) divide by (log(x) divide by log(3))
Similar expressions
(2*x-3)/(log(x)/log(3))

The derivative of the function/ (2*x+3)/(log(x)/log(3))

Derivative of (2*x+3)/(log(x)/log(3))

The solution

You have entered [src]

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

$$\frac{2 x + 3}{\frac{1}{\log{\left(3 \right)}} \log{\left(x \right)}}$$

d /2*x + 3 \
--|--------|
dx|/log(x)\|
  ||------||
  \\log(3)//

$$\frac{d}{d x} \frac{2 x + 3}{\frac{1}{\log{\left(3 \right)}} \log{\left(x \right)}}$$

Transform

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \left(2 x + 3\right) \log{\left(3 \right)}$ and $g{\left(x \right)} = \log{\left(x \right)}$ .

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.
  1. Differentiate $2 x + 3$ term by term:
    1. The derivative of the constant $3$ is zero.
    2. 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: $2$
    The result is: $2$
  So, the result is: $2 \log{\left(3 \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
Now plug in to the quotient rule:

$\frac{2 \log{\left(3 \right)} \log{\left(x \right)} - \frac{\left(2 x + 3\right) \log{\left(3 \right)}}{x}}{\log{\left(x \right)}^{2}}$
Now simplify:

$\frac{\left(2 x \log{\left(x \right)} - 2 x - 3\right) \log{\left(3 \right)}}{x \log{\left(x \right)}^{2}}$

The answer is:

$\frac{\left(2 x \log{\left(x \right)} - 2 x - 3\right) \log{\left(3 \right)}}{x \log{\left(x \right)}^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

/     /      2   \          \       
|     |1 + ------|*(3 + 2*x)|       
|     \    log(x)/          |       
|-4 + ----------------------|*log(3)
\               x           /       
------------------------------------
                  2                 
             x*log (x)

$$\frac{\left(-4 + \frac{\left(1 + \frac{2}{\log{\left(x \right)}}\right) \left(2 x + 3\right)}{x}\right) \log{\left(3 \right)}}{x \log{\left(x \right)}^{2}}$$

Simplify

The third derivative [src]

  /                       /      3         3   \\       
  |             (3 + 2*x)*|1 + ------ + -------||       
  |                       |    log(x)      2   ||       
  |      6                \             log (x)/|       
2*|3 + ------ - --------------------------------|*log(3)
  \    log(x)                  x                /       
--------------------------------------------------------
                        2    2                          
                       x *log (x)

$$\frac{2 \cdot \left(3 + \frac{6}{\log{\left(x \right)}} - \frac{\left(2 x + 3\right) \left(1 + \frac{3}{\log{\left(x \right)}} + \frac{3}{\log{\left(x \right)}^{2}}\right)}{x}\right) \log{\left(3 \right)}}{x^{2} \log{\left(x \right)}^{2}}$$

Simplify

The graph

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Derivative of (2*x+3)/(log(x)/log(3))

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Piecewise:

The solution