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Similar expressions
y=ln(3x^2−2x-5)

The derivative of the function/ y=ln(3x^2−2x+5)

Derivative of y=ln(3x^2−2x+5)

The solution

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   /   2          \
log\3*x  - 2*x + 5/

$$\log{\left(3 x^{2} - 2 x + 5 \right)}$$

d /   /   2          \\
--\log\3*x  - 2*x + 5//
dx

$$\frac{d}{d x} \log{\left(3 x^{2} - 2 x + 5 \right)}$$

Transform

Detail solution

Let $u = 3 x^{2} - 2 x + 5$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(3 x^{2} - 2 x + 5\right)$ :
1. Differentiate $3 x^{2} - 2 x + 5$ 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: $6 x$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    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: $2$
    So, the result is: $-2$
  3. The derivative of the constant $5$ is zero.
  The result is: $6 x - 2$
The result of the chain rule is:

$\frac{6 x - 2}{3 x^{2} - 2 x + 5}$
Now simplify:

$\frac{2 \cdot \left(3 x - 1\right)}{3 x^{2} - 2 x + 5}$

The answer is:

$\frac{2 \cdot \left(3 x - 1\right)}{3 x^{2} - 2 x + 5}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   -2 + 6*x   
--------------
   2          
3*x  - 2*x + 5

$$\frac{6 x - 2}{3 x^{2} - 2 x + 5}$$

Simplify

The second derivative [src]

  /                2 \
  |    2*(-1 + 3*x)  |
2*|3 - --------------|
  |                 2|
  \    5 - 2*x + 3*x /
----------------------
                 2    
    5 - 2*x + 3*x

$$\frac{2 \left(- \frac{2 \left(3 x - 1\right)^{2}}{3 x^{2} - 2 x + 5} + 3\right)}{3 x^{2} - 2 x + 5}$$

Simplify

The third derivative [src]

             /                 2 \
             |     4*(-1 + 3*x)  |
4*(-1 + 3*x)*|-9 + --------------|
             |                  2|
             \     5 - 2*x + 3*x /
----------------------------------
                        2         
        /             2\          
        \5 - 2*x + 3*x /

$$\frac{4 \cdot \left(3 x - 1\right) \left(\frac{4 \left(3 x - 1\right)^{2}}{3 x^{2} - 2 x + 5} - 9\right)}{\left(3 x^{2} - 2 x + 5\right)^{2}}$$

Simplify

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Derivative of y=ln(3x^2−2x+5)

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The solution