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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)

Function f() - derivative -N order at the point
v

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The solution

You have entered [src] 
   /   2          \
log\2*x  - 3*x + 6/
log⁡((2x2−3x)+6)\log{\left(\left(2 x^{2} - 3 x\right) + 6 \right)} 
log(2*x^2 - 3*x + 6)
Detail solution

  1. Let u=(2x2−3x)+6u = \left(2 x^{2} - 3 x\right) + 6.

  2. The derivative of log⁡(u)\log{\left(u \right)} is 1u\frac{1}{u}.

  3. Then, apply the chain rule. Multiply by ddx((2x2−3x)+6)\frac{d}{d x} \left(\left(2 x^{2} - 3 x\right) + 6\right):

    1. Differentiate (2x2−3x)+6\left(2 x^{2} - 3 x\right) + 6 term by term:

      1. Differentiate 2x2−3x2 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: x2x^{2} goes to 2x2 x

          So, the result is: 4x4 x

        2. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: xx goes to 11

          So, the result is: −3-3

        The result is: 4x−34 x - 3

      2. The derivative of the constant 66 is zero.

      The result is: 4x−34 x - 3

    The result of the chain rule is:

    4x−3(2x2−3x)+6\frac{4 x - 3}{\left(2 x^{2} - 3 x\right) + 6}

  4. Now simplify:

    4x−32x2−3x+6\frac{4 x - 3}{2 x^{2} - 3 x + 6}

The answer is:

4x−32x2−3x+6\frac{4 x - 3}{2 x^{2} - 3 x + 6}

The graph
02468-8-6-4-2-1010-510
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   -3 + 4*x   
--------------
   2          
2*x  - 3*x + 6
4x−3(2x2−3x)+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
−(4x−3)22x2−3x+6+42x2−3x+6\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 /
2(4x−3)((4x−3)22x2−3x+6−6)(2x2−3x+6)2\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 /
2(4x−3)((4x−3)22x2−3x+6−6)(2x2−3x+6)2\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
Derivative of y=log(2x^2-3x+6)
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