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y=ln(x²-4x+4)
The derivative of the function/ y=ln(x²-4x+4)

Derivative of y=ln(x²-4x+4)

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
   / 2          \
log\x  - 4*x + 4/
log⁡((x2−4x)+4)\log{\left(\left(x^{2} - 4 x\right) + 4 \right)} 
log(x^2 - 4*x + 4)
Detail solution

  1. Let u=(x2−4x)+4u = \left(x^{2} - 4 x\right) + 4.

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

  3. Then, apply the chain rule. Multiply by ddx((x2−4x)+4)\frac{d}{d x} \left(\left(x^{2} - 4 x\right) + 4\right):

    1. Differentiate (x2−4x)+4\left(x^{2} - 4 x\right) + 4 term by term:

      1. Differentiate x2−4xx^{2} - 4 x term by term:

        1. Apply the power rule: x2x^{2} goes to 2x2 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: −4-4

        The result is: 2x−42 x - 4

      2. The derivative of the constant 44 is zero.

      The result is: 2x−42 x - 4

    The result of the chain rule is:

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

  4. Now simplify:

    2x−2\frac{2}{x - 2}

The answer is:

2x−2\frac{2}{x - 2}

The graph
02468-8-6-4-2-1010-5050
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
  -4 + 2*x  
------------
 2          
x  - 4*x + 4
2x−4(x2−4x)+4\frac{2 x - 4}{\left(x^{2} - 4 x\right) + 4}
Simplify
The second derivative [src] 
  /              2 \
  |    2*(-2 + x)  |
2*|1 - ------------|
  |         2      |
  \    4 + x  - 4*x/
--------------------
         2          
    4 + x  - 4*x
2(−2(x−2)2x2−4x+4+1)x2−4x+4\frac{2 \left(- \frac{2 \left(x - 2\right)^{2}}{x^{2} - 4 x + 4} + 1\right)}{x^{2} - 4 x + 4}
Simplify
The third derivative [src] 
  /               2 \         
  |     4*(-2 + x)  |         
4*|-3 + ------------|*(-2 + x)
  |          2      |         
  \     4 + x  - 4*x/         
------------------------------
                     2        
       /     2      \         
       \4 + x  - 4*x/
4(x−2)(4(x−2)2x2−4x+4−3)(x2−4x+4)2\frac{4 \left(x - 2\right) \left(\frac{4 \left(x - 2\right)^{2}}{x^{2} - 4 x + 4} - 3\right)}{\left(x^{2} - 4 x + 4\right)^{2}}
Simplify
The graph
Derivative of y=ln(x²-4x+4)
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