Calculus Examples

$x^{3} - 6 x^{2} + 9 x$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $x^{3} - 6 x^{2} + 9 x$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [9 x]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [9 x]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [9 x]$

Step 2

Evaluate $\frac{d}{d x} [- 6 x^{2}]$ .

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Step 2.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x^{2}$ with respect to $x$ is $- 6 \frac{d}{d x} [x^{2}]$ .

$3 x^{2} - 6 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [9 x]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$3 x^{2} - 6 (2 x) + \frac{d}{d x} [9 x]$

Step 2.3

Multiply $2$ by $- 6$ .

$3 x^{2} - 12 x + \frac{d}{d x} [9 x]$

Step 3

Evaluate $\frac{d}{d x} [9 x]$ .

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Step 3.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x$ with respect to $x$ is $9 \frac{d}{d x} [x]$ .

$3 x^{2} - 12 x + 9 \frac{d}{d x} [x]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$3 x^{2} - 12 x + 9 \cdot 1$

Step 3.3

Multiply $9$ by $1$ .

$3 x^{2} - 12 x + 9$