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Identical expressions
y=(lnx+2x)(three x^3-17x)
y equally (lnx plus 2x)(3x cubed minus 17x)
y equally (lnx plus 2x)(three x cubed minus 17x)
y=(lnx+2x)(3x3-17x)
y=lnx+2x3x3-17x
y=(lnx+2x)(3x³-17x)
y=(lnx+2x)(3x to the power of 3-17x)
y=lnx+2x3x^3-17x
Similar expressions
y=(lnx+2x)(3x^3+17x)
y=(lnx-2x)(3x^3-17x)

The derivative of the function/ y=(lnx+2x)(3x^3-17x)

Derivative of y=(lnx+2x)(3x^3-17x)

The solution

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               /   3       \
(log(x) + 2*x)*\3*x  - 17*x/

$$\left(2 x + \log{\left(x \right)}\right) \left(3 x^{3} - 17 x\right)$$

(log(x) + 2*x)*(3*x^3 - 17*x)

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = 2 x + \log{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $2 x + \log{\left(x \right)}$ term by term:
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  2. 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: $2$
  The result is: $2 + \frac{1}{x}$
$g{\left(x \right)} = 3 x^{3} - 17 x$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $3 x^{3} - 17 x$ 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: $9 x^{2}$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $-17$
  The result is: $9 x^{2} - 17$
The result is: $\left(2 + \frac{1}{x}\right) \left(3 x^{3} - 17 x\right) + \left(2 x + \log{\left(x \right)}\right) \left(9 x^{2} - 17\right)$
Now simplify:

$\left(2 x + 1\right) \left(3 x^{2} - 17\right) + \left(2 x + \log{\left(x \right)}\right) \left(9 x^{2} - 17\right)$

The answer is:

$\left(2 x + 1\right) \left(3 x^{2} - 17\right) + \left(2 x + \log{\left(x \right)}\right) \left(9 x^{2} - 17\right)$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

/         2\                  /    1\ /   3       \
\-17 + 9*x /*(log(x) + 2*x) + |2 + -|*\3*x  - 17*x/
                              \    x/

$$\left(2 + \frac{1}{x}\right) \left(3 x^{3} - 17 x\right) + \left(2 x + \log{\left(x \right)}\right) \left(9 x^{2} - 17\right)$$

Simplify

The second derivative [src]

           2                                               
  -17 + 3*x      /         2\ /    1\                      
- ---------- + 2*\-17 + 9*x /*|2 + -| + 18*x*(2*x + log(x))
      x                       \    x/

$$18 x \left(2 x + \log{\left(x \right)}\right) + 2 \left(2 + \frac{1}{x}\right) \left(9 x^{2} - 17\right) - \frac{3 x^{2} - 17}{x}$$

Simplify

The third derivative [src]

                     /         2\     /         2\               
                   3*\-17 + 9*x /   2*\-17 + 3*x /        /    1\
18*log(x) + 36*x - -------------- + -------------- + 54*x*|2 + -|
                          2                2              \    x/
                         x                x

$$54 x \left(2 + \frac{1}{x}\right) + 36 x + 18 \log{\left(x \right)} + \frac{2 \left(3 x^{2} - 17\right)}{x^{2}} - \frac{3 \left(9 x^{2} - 17\right)}{x^{2}}$$

Simplify

The graph

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Derivative of y=(lnx+2x)(3x^3-17x)

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The solution