Derivative of y=ln((3x-2)(4-x))

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log((3*x - 2)*(4 - x))

$$\log{\left(\left(4 - x\right) \left(3 x - 2\right) \right)}$$

log((3*x - 2)*(4 - x))

Detail solution

Let $u = \left(4 - x\right) \left(3 x - 2\right)$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(4 - x\right) \left(3 x - 2\right)$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = 3 x - 2$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Differentiate $3 x - 2$ 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: $x$ goes to $1$
      So, the result is: $3$
    2. The derivative of the constant $-2$ is zero.
    The result is: $3$
  $g{\left(x \right)} = 4 - x$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $4 - x$ term by term:
    1. The derivative of the constant $4$ is zero.
    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: $-1$
    The result is: $-1$
  The result is: $14 - 6 x$
The result of the chain rule is:

$\frac{14 - 6 x}{\left(4 - x\right) \left(3 x - 2\right)}$
Now simplify:

$\frac{2 \left(3 x - 7\right)}{\left(x - 4\right) \left(3 x - 2\right)}$

The answer is:

$\frac{2 \left(3 x - 7\right)}{\left(x - 4\right) \left(3 x - 2\right)}$

The graph

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The first derivative [src]

     14 - 6*x    
-----------------
(4 - x)*(3*x - 2)

$$\frac{14 - 6 x}{\left(4 - x\right) \left(3 x - 2\right)}$$

Simplify

The second derivative [src]

  /    -7 + 3*x   3*(-7 + 3*x)\
2*|3 - -------- - ------------|
  \     -4 + x      -2 + 3*x  /
-------------------------------
      (-4 + x)*(-2 + 3*x)

$$\frac{2 \left(- \frac{3 \left(3 x - 7\right)}{3 x - 2} + 3 - \frac{3 x - 7}{x - 4}\right)}{\left(x - 4\right) \left(3 x - 2\right)}$$

Simplify

The third derivative [src]

  /     9         3       -7 + 3*x   9*(-7 + 3*x)       3*(-7 + 3*x)   \
4*|- -------- - ------ + --------- + ------------ + -------------------|
  |  -2 + 3*x   -4 + x           2             2    (-4 + x)*(-2 + 3*x)|
  \                      (-4 + x)    (-2 + 3*x)                        /
------------------------------------------------------------------------
                          (-4 + x)*(-2 + 3*x)

$$\frac{4 \left(\frac{9 \left(3 x - 7\right)}{\left(3 x - 2\right)^{2}} - \frac{9}{3 x - 2} + \frac{3 \left(3 x - 7\right)}{\left(x - 4\right) \left(3 x - 2\right)} - \frac{3}{x - 4} + \frac{3 x - 7}{\left(x - 4\right)^{2}}\right)}{\left(x - 4\right) \left(3 x - 2\right)}$$

Simplify

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Derivative of y=ln((3x-2)(4-x))

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