Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of (4x-3)^2
Derivative of 8lnx
Derivative of (x+5)*sqrt(x)
Derivative of (x^2+1)^4
Identical expressions
y=log three (x+ two)/2x+3
y equally logarithm of 3(x plus 2) divide by 2x plus 3
y equally logarithm of three (x plus two) divide by 2x plus 3
y=log3x+2/2x+3
y=log3(x+2) divide by 2x+3
Similar expressions
y=log3(x+2)/2x-3
y=log3(x+2)/(2x+3)
y=log3(x-2)/2x+3

The derivative of the function/ y=log3(x+2)/2x+3

Derivative of y=log3(x+2)/2x+3

The solution

You have entered [src]

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

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

Transform

Detail solution

Differentiate $x \frac{\frac{1}{\log{\left(3 \right)}} \log{\left(x + 2 \right)}}{2} + 3$ term by term:
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = x \log{\left(x + 2 \right)}$ and $g{\left(x \right)} = 2 \log{\left(3 \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the product rule:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Apply the power rule: $x$ goes to $1$
    $g{\left(x \right)} = \log{\left(x + 2 \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = x + 2$ .
    2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 2\right)$ :
      1. Differentiate $x + 2$ term by term:
        
        Apply the power rule: $x$ goes to $1$
        
        The derivative of the constant $2$ is zero.
        
        The result is: $1$
      The result of the chain rule is:
      
      $\frac{1}{x + 2}$
    The result is: $\frac{x}{x + 2} + \log{\left(x + 2 \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of the constant $2 \log{\left(3 \right)}$ is zero.
  Now plug in to the quotient rule:
  
  $\frac{\frac{x}{x + 2} + \log{\left(x + 2 \right)}}{2 \log{\left(3 \right)}}$
2. The derivative of the constant $3$ is zero.
The result is: $\frac{\frac{x}{x + 2} + \log{\left(x + 2 \right)}}{2 \log{\left(3 \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

        x     
1 - --------- 
    2*(2 + x) 
--------------
(2 + x)*log(3)

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of y=log3(x+2)/2x+3

The graph:

Piecewise:

The solution