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

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

Function f() - derivative -N order at the point
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from to

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The solution

You have entered [src] 
   /   3      2\
log\2*x  + 3*x /
log⁡(2x3+3x2)\log{\left(2 x^{3} + 3 x^{2} \right)} 
d /   /   3      2\\
--\log\2*x  + 3*x //
dx
ddxlog⁡(2x3+3x2)\frac{d}{d x} \log{\left(2 x^{3} + 3 x^{2} \right)}
Transform
Detail solution

  1. Let u=2x3+3x2u = 2 x^{3} + 3 x^{2}.

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

  3. Then, apply the chain rule. Multiply by ddx(2x3+3x2)\frac{d}{d x} \left(2 x^{3} + 3 x^{2}\right):

    1. Differentiate 2x3+3x22 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: x3x^{3} goes to 3x23 x^{2}

        So, the result is: 6x26 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: x2x^{2} goes to 2x2 x

        So, the result is: 6x6 x

      The result is: 6x2+6x6 x^{2} + 6 x

    The result of the chain rule is:

    6x2+6x2x3+3x2\frac{6 x^{2} + 6 x}{2 x^{3} + 3 x^{2}}

  4. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-100100
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
          2
 6*x + 6*x 
-----------
   3      2
2*x  + 3*x
6x2+6x2x3+3x2\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)
6⋅(2x−6(x+1)22x+3+1)x2⋅(2x+3)\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)
12⋅(1+36(x+1)3x(2x+3)2−9(x+1)(2x+1)x(2x+3))x2⋅(2x+3)\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
Derivative of ln(2x^3+3x^2)
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