Derivative of `x^3-3*x`

Step-by-step differentiation

Answer

\frac{d}{d x} [x^{3} - 3 x]

3 \cdot x^{2} - 3

Step-by-step solution

1Difference Rule
$\frac{d}{d x} [x^{3} - 3 x] = \frac{d}{d x} [x^{3}] - \frac{d}{d x} [3 x]$
Use the difference rule: differentiate each term on its own, then subtract.
2Power Rule
$\frac{d}{d x} [x^{3}] = 3 \cdot x^{3 - 1}$
The exponent is $3$ . Bring it down in front as a coefficient, then reduce the power by $1$ .
3Constant Multiple Rule
$\frac{d}{d x} [3 x] = 3 \cdot \frac{d}{d x} [x]$
The factor $3$ is constant — pull it outside so only the variable part needs differentiating.
4Power Rule
$\frac{d}{d x} [x] = 1$
$x$ is $x^{1}$ : by the power rule, the exponent $1$ comes down and the power drops to $x^{0} = 1$ , so the derivative is $1$ .
=Combine
$\frac{d}{d x} [3 x] = 3 \cdot 1 = 3$
Multiply: $3 \cdot 1 = 3$ .
=Combine
$\frac{d}{d x} [x^{3} - 3 x] = 3 \cdot x^{2} - 3 = 3 \cdot x^{2} - 3$
Both terms are differentiated. Subtract: $3 \cdot x^{2} - 3 = 3 \cdot x^{2} - 3$ .
✓Final Answer
$\frac{d}{d x} [x^{3} - 3 x] = 3 \cdot x^{2} - 3$
The first derivative simplifies to $3 \cdot x^{2} - 3$ .

Understanding this derivative

Differentiating x³-3x exercises 3 distinct techniques — the power rule, the sum and difference rules and the constant multiple rule. Problems like this one are useful practice precisely because the rules have to be combined in the right order rather than applied in isolation.

The power rule is the workhorse step: bring the exponent down as a coefficient and reduce the exponent by one. Sums and differences differentiate term by term, so the expression splits into independent pieces that are each handled with their own rule. Constant coefficients pass straight through differentiation — they are carried along unchanged while the variable part is differentiated.

A good way to verify a derivative by hand is to compare it against the step-by-step breakdown above — each step names the rule being applied, so you can pinpoint exactly where your own work diverges if the answers differ.

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Derivative of x^3-3*x

Step-by-step solution

Understanding this derivative

Derivative of `x^3-3*x`