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y=x*(2+3*lnx)+2x³

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Identical expressions
y=x*(two + three *lnx)+2x³
y equally x multiply by (2 plus 3 multiply by lnx) plus 2x³
y equally x multiply by (two plus three multiply by lnx) plus 2x³
y=x(2+3lnx)+2x³
y=x2+3lnx+2x³
Similar expressions
y=x*(2+3*lnx)-2x³
y=x*(2-3*lnx)+2x³

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

Derivative of y=x(2+3lnx)+2x³

The solution

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                      3
x*(2 + 3*log(x)) + 2*x

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

x*(2 + 3*log(x)) + 2*x^3

Detail solution

Differentiate $2 x^{3} + x \left(3 \log{\left(x \right)} + 2\right)$ term by term:
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = 3 \log{\left(x \right)} + 2$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $3 \log{\left(x \right)} + 2$ term by term:
    1. The derivative of the constant $2$ is zero.
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
      So, the result is: $\frac{3}{x}$
    The result is: $\frac{3}{x}$
  The result is: $3 \log{\left(x \right)} + 5$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
  So, the result is: $6 x^{2}$
The result is: $6 x^{2} + 3 \log{\left(x \right)} + 5$

The answer is:

$6 x^{2} + 3 \log{\left(x \right)} + 5$

The graph

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The first derivative [src]

                  2
5 + 3*log(x) + 6*x

$$6 x^{2} + 3 \log{\left(x \right)} + 5$$

Simplify

The second derivative [src]

  /1      \
3*|- + 4*x|
  \x      /

$$3 \left(4 x + \frac{1}{x}\right)$$

Simplify

The third derivative [src]

  /    1 \
3*|4 - --|
  |     2|
  \    x /

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

Simplify

The graph

Derivative of y=x*(2+3*lnx)+2x³