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y=x*(2+3*lnx)+2x³
The derivative of the function/ y=x*(2+3*lnx)+2x³

Derivative of y=x*(2+3*lnx)+2x³

Function f() - derivative -N order at the point
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The solution

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                      3
x*(2 + 3*log(x)) + 2*x
2x3+x(3log⁡(x)+2)2 x^{3} + x \left(3 \log{\left(x \right)} + 2\right) 
x*(2 + 3*log(x)) + 2*x^3
Detail solution

  1. Differentiate 2x3+x(3log⁡(x)+2)2 x^{3} + x \left(3 \log{\left(x \right)} + 2\right) term by term:

    1. Apply the product rule:

      ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\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(x)=xf{\left(x \right)} = x; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Apply the power rule: xx goes to 11

      g(x)=3log⁡(x)+2g{\left(x \right)} = 3 \log{\left(x \right)} + 2; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. Differentiate 3log⁡(x)+23 \log{\left(x \right)} + 2 term by term:

        1. The derivative of the constant 22 is zero.

        2. The derivative of a constant times a function is the constant times the derivative of the function.

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

          So, the result is: 3x\frac{3}{x}

        The result is: 3x\frac{3}{x}

      The result is: 3log⁡(x)+53 \log{\left(x \right)} + 5

    2. 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}

    The result is: 6x2+3log⁡(x)+56 x^{2} + 3 \log{\left(x \right)} + 5

The answer is:

6x2+3log⁡(x)+56 x^{2} + 3 \log{\left(x \right)} + 5

The graph
02468-8-6-4-2-1010-25002500
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
                  2
5 + 3*log(x) + 6*x
6x2+3log⁡(x)+56 x^{2} + 3 \log{\left(x \right)} + 5
Simplify
The second derivative [src] 
  /1      \
3*|- + 4*x|
  \x      /
3(4x+1x)3 \left(4 x + \frac{1}{x}\right)
Simplify
The third derivative [src] 
  /    1 \
3*|4 - --|
  |     2|
  \    x /
3(4−1x2)3 \left(4 - \frac{1}{x^{2}}\right)
Simplify
The graph
Derivative of y=x*(2+3*lnx)+2x³
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