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Derivative of:
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Identical expressions
(x^ three -x)(x^ two + two)
(x cubed minus x)(x squared plus 2)
(x to the power of three minus x)(x to the power of two plus two)
(x3-x)(x2+2)
x3-xx2+2
(x³-x)(x²+2)
(x to the power of 3-x)(x to the power of 2+2)
x^3-xx^2+2
Similar expressions
(x^3-x)(x^2-2)
(x^3+x)(x^2+2)

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

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

The solution

You have entered [src]

/ 3    \ / 2    \
\x  - x/*\x  + 2/

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

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

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

Transform

Detail solution

Apply the product rule:

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

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

$5 x^{4} + 3 x^{2} - 2$

The answer is:

$5 x^{4} + 3 x^{2} - 2$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

/        2\ / 2    \       / 3    \
\-1 + 3*x /*\x  + 2/ + 2*x*\x  - x/

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

Simplify

The second derivative [src]

    /        2\
2*x*\3 + 10*x /

$$2 x \left(10 x^{2} + 3\right)$$

Simplify

The third derivative [src]

  /        2\
6*\1 + 10*x /

$$6 \cdot \left(10 x^{2} + 1\right)$$

Simplify

The graph

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