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Derivative of:
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Identical expressions
x^ three *(x+ two)^ two
x cubed multiply by (x plus 2) squared
x to the power of three multiply by (x plus two) to the power of two
x3*(x+2)2
x3*x+22
x³*(x+2)²
x to the power of 3*(x+2) to the power of 2
x^3(x+2)^2
x3(x+2)2
x3x+22
x^3x+2^2
Similar expressions
x^3*(x-2)^2

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

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

The solution

You have entered [src]

 3        2
x *(x + 2)

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

d / 3        2\
--\x *(x + 2) /
dx

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

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

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

The answer is:

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

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 *(x + 2)

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

Simplify

The second derivative [src]

    / 2            2              \
2*x*\x  + 3*(2 + x)  + 6*x*(2 + x)/

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

Simplify

The third derivative [src]

  /       2      2              \
6*\(2 + x)  + 3*x  + 6*x*(2 + x)/

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

Simplify

The graph

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