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Derivative of y=log(2x^2-3x+6)
Identical expressions
y=log(two x^2-3x+ six)
y equally logarithm of (2x squared minus 3x plus 6)
y equally logarithm of (two x squared minus 3x plus six)
y=log(2x2-3x+6)
y=log2x2-3x+6
y=log(2x²-3x+6)
y=log(2x to the power of 2-3x+6)
y=log2x^2-3x+6
Similar expressions
y=log(2x^2-3x-6)
y=log(2x^2+3x+6)

The derivative of the function/ y=log(2x^2-3x+6)

Derivative of y=log(2x^2-3x+6)

The solution

You have entered [src]

   /   2          \
log\2*x  - 3*x + 6/

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

log(2*x^2 - 3*x + 6)

Detail solution

Let $u = \left(2 x^{2} - 3 x\right) + 6$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\left(2 x^{2} - 3 x\right) + 6\right)$ :
1. Differentiate $\left(2 x^{2} - 3 x\right) + 6$ term by term:
  1. Differentiate $2 x^{2} - 3 x$ term by term:
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x^{2}$ goes to $2 x$
      So, the result is: $4 x$
    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: $-3$
    The result is: $4 x - 3$
  2. The derivative of the constant $6$ is zero.
  The result is: $4 x - 3$
The result of the chain rule is:

$\frac{4 x - 3}{\left(2 x^{2} - 3 x\right) + 6}$
Now simplify:

$\frac{4 x - 3}{2 x^{2} - 3 x + 6}$

The answer is:

$\frac{4 x - 3}{2 x^{2} - 3 x + 6}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   -3 + 4*x   
--------------
   2          
2*x  - 3*x + 6

$$\frac{4 x - 3}{\left(2 x^{2} - 3 x\right) + 6}$$

Simplify

The second derivative [src]

               2  
     (-3 + 4*x)   
4 - --------------
                 2
    6 - 3*x + 2*x 
------------------
               2  
  6 - 3*x + 2*x

$$\frac{- \frac{\left(4 x - 3\right)^{2}}{2 x^{2} - 3 x + 6} + 4}{2 x^{2} - 3 x + 6}$$

Simplify

The third derivative [src]

  /                2  \           
  |      (-3 + 4*x)   |           
2*|-6 + --------------|*(-3 + 4*x)
  |                  2|           
  \     6 - 3*x + 2*x /           
----------------------------------
                        2         
        /             2\          
        \6 - 3*x + 2*x /

$$\frac{2 \left(4 x - 3\right) \left(\frac{\left(4 x - 3\right)^{2}}{2 x^{2} - 3 x + 6} - 6\right)}{\left(2 x^{2} - 3 x + 6\right)^{2}}$$

Simplify

3-я производная [src]

  /                2  \           
  |      (-3 + 4*x)   |           
2*|-6 + --------------|*(-3 + 4*x)
  |                  2|           
  \     6 - 3*x + 2*x /           
----------------------------------
                        2         
        /             2\          
        \6 - 3*x + 2*x /

$$\frac{2 \left(4 x - 3\right) \left(\frac{\left(4 x - 3\right)^{2}}{2 x^{2} - 3 x + 6} - 6\right)}{\left(2 x^{2} - 3 x + 6\right)^{2}}$$

Simplify

The graph

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Derivative of y=log(2x^2-3x+6)

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The solution