Derivative of y=x³ln(x²+4x)

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 3    / 2      \
x *log\x  + 4*x/

$$x^{3} \log{\left(x^{2} + 4 x \right)}$$

d / 3    / 2      \\
--\x *log\x  + 4*x//
dx

$$\frac{d}{d x} x^{3} \log{\left(x^{2} + 4 x \right)}$$

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x^{3}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
$g{\left(x \right)} = \log{\left(x^{2} + 4 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x^{2} + 4 x$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} + 4 x\right)$ :
  1. Differentiate $x^{2} + 4 x$ term by term:
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    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: $4$
    The result is: $2 x + 4$
  The result of the chain rule is:
  
  $\frac{2 x + 4}{x^{2} + 4 x}$
The result is: $\frac{x^{3} \cdot \left(2 x + 4\right)}{x^{2} + 4 x} + 3 x^{2} \log{\left(x^{2} + 4 x \right)}$
Now simplify:

$\frac{x^{2} \cdot \left(2 x + 3 \left(x + 4\right) \log{\left(x \left(x + 4\right) \right)} + 4\right)}{x + 4}$

The answer is:

$\frac{x^{2} \cdot \left(2 x + 3 \left(x + 4\right) \log{\left(x \left(x + 4\right) \right)} + 4\right)}{x + 4}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                      3          
   2    / 2      \   x *(4 + 2*x)
3*x *log\x  + 4*x/ + ------------
                        2        
                       x  + 4*x

$$\frac{x^{3} \cdot \left(2 x + 4\right)}{x^{2} + 4 x} + 3 x^{2} \log{\left(x^{2} + 4 x \right)}$$

Simplify

The second derivative [src]

    /                                 /             2\\
    |                                 |    2*(2 + x) ||
    |                               x*|1 - ----------||
    |                   6*(2 + x)     \    x*(4 + x) /|
2*x*|3*log(x*(4 + x)) + --------- + ------------------|
    \                     4 + x           4 + x       /

$$2 x \left(\frac{x \left(1 - \frac{2 \left(x + 2\right)^{2}}{x \left(x + 4\right)}\right)}{x + 4} + 3 \log{\left(x \left(x + 4\right) \right)} + \frac{6 \left(x + 2\right)}{x + 4}\right)$$

Simplify

The third derivative [src]

  /                                    /             2\               /             2\\
  |                                    |    2*(2 + x) |               |    4*(2 + x) ||
  |                                9*x*|1 - ----------|   2*x*(2 + x)*|3 - ----------||
  |                   18*(2 + x)       \    x*(4 + x) /               \    x*(4 + x) /|
2*|3*log(x*(4 + x)) + ---------- + -------------------- - ----------------------------|
  |                     4 + x             4 + x                            2          |
  \                                                                 (4 + x)           /

$$2 \cdot \left(\frac{9 x \left(1 - \frac{2 \left(x + 2\right)^{2}}{x \left(x + 4\right)}\right)}{x + 4} - \frac{2 x \left(3 - \frac{4 \left(x + 2\right)^{2}}{x \left(x + 4\right)}\right) \left(x + 2\right)}{\left(x + 4\right)^{2}} + 3 \log{\left(x \left(x + 4\right) \right)} + \frac{18 \left(x + 2\right)}{x + 4}\right)$$

Simplify

The graph

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Derivative of y=x³ln(x²+4x)

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The solution