Calculus Examples

$f (x) = x^{3} - \frac{3}{2} x^{2}$

Step 1

Differentiate.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Evaluate $\frac{d}{d x} [- \frac{3}{2} x^{2}]$ .

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Step 2.1

Since $- \frac{3}{2}$ is constant with respect to $x$ , the derivative of $- \frac{3}{2} x^{2}$ with respect to $x$ is $- \frac{3}{2} \frac{d}{d x} [x^{2}]$ .

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

Step 2.2

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

$3 x^{2} - \frac{3}{2} (2 x)$

Step 2.3

Multiply $2$ by $- 1$ .

$3 x^{2} - 2 (\frac{3}{2}) x$

Step 2.4

Combine $- 2$ and $\frac{3}{2}$ .

$3 x^{2} + \frac{- 2 \cdot 3}{2} x$

Step 2.5

Multiply $- 2$ by $3$ .

$3 x^{2} + \frac{- 6}{2} x$

Step 2.6

Combine $\frac{- 6}{2}$ and $x$ .

$3 x^{2} + \frac{- 6 x}{2}$

Step 2.7

Cancel the common factor of $- 6$ and $2$ .

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Step 2.7.1

Factor $2$ out of $- 6 x$ .

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

Step 2.7.2

Cancel the common factors.

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Step 2.7.2.1

Factor $2$ out of $2$ .

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

Step 2.7.2.2

Cancel the common factor.

$3 x^{2} + \frac{2 (- 3 x)}{2 \cdot 1}$

Step 2.7.2.3

Rewrite the expression.

$3 x^{2} + \frac{- 3 x}{1}$

Step 2.7.2.4

Divide $- 3 x$ by $1$ .

$3 x^{2} - 3 x$