Derivative of y=ln(x²-4x+4)

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   / 2          \
log\x  - 4*x + 4/

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

log(x^2 - 4*x + 4)

Detail solution

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

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

$\frac{2}{x - 2}$

The answer is:

$\frac{2}{x - 2}$

The graph

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Plot the graph f'(x)

The first derivative [src]

  -4 + 2*x  
------------
 2          
x  - 4*x + 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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of y=ln(x²-4x+4)

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The solution