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ln(two x^ three +3x^2)
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ln2x^3+3x^2
Similar expressions
ln(2x^3-3x^2)

The derivative of the function/ ln(2x^3+3x^2)

Derivative of ln(2x^3+3x^2)

The solution

You have entered [src]

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

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

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

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

Transform

Detail solution

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

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

$\frac{6 \left(x + 1\right)}{x \left(2 x + 3\right)}$

The answer is:

$\frac{6 \left(x + 1\right)}{x \left(2 x + 3\right)}$

The graph

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Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /                   2\
  |          6*(1 + x) |
6*|1 + 2*x - ----------|
  \           3 + 2*x  /
------------------------
       2                
      x *(3 + 2*x)

$$\frac{6 \cdot \left(2 x - \frac{6 \left(x + 1\right)^{2}}{2 x + 3} + 1\right)}{x^{2} \cdot \left(2 x + 3\right)}$$

Simplify

The third derivative [src]

   /              3                       \
   |    36*(1 + x)     9*(1 + x)*(1 + 2*x)|
12*|1 + ------------ - -------------------|
   |               2       x*(3 + 2*x)    |
   \    x*(3 + 2*x)                       /
-------------------------------------------
                 2                         
                x *(3 + 2*x)

$$\frac{12 \cdot \left(1 + \frac{36 \left(x + 1\right)^{3}}{x \left(2 x + 3\right)^{2}} - \frac{9 \left(x + 1\right) \left(2 x + 1\right)}{x \left(2 x + 3\right)}\right)}{x^{2} \cdot \left(2 x + 3\right)}$$

Simplify

The graph

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Derivative of ln(2x^3+3x^2)

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The solution