Calculus Examples

$y = \frac{3 x^{2}}{2} + 1$

Step 1

By the Sum Rule, the derivative of $\frac{3 x^{2}}{2} + 1$ with respect to $x$ is $\frac{d}{d x} [\frac{3 x^{2}}{2}] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [\frac{3 x^{2}}{2}] + \frac{d}{d x} [1]$

Step 2

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

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Step 2.1

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

$\frac{3}{2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$

Step 2.2

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

$\frac{3}{2} (2 x) + \frac{d}{d x} [1]$

Step 2.3

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

$\frac{2 \cdot 3}{2} x + \frac{d}{d x} [1]$

Step 2.4

Multiply $2$ by $3$ .

$\frac{6}{2} x + \frac{d}{d x} [1]$

Step 2.5

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

$\frac{6 x}{2} + \frac{d}{d x} [1]$

Step 2.6

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

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Step 2.6.1

Factor $2$ out of $6 x$ .

$\frac{2 (3 x)}{2} + \frac{d}{d x} [1]$

Step 2.6.2

Cancel the common factors.

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Step 2.6.2.1

Factor $2$ out of $2$ .

$\frac{2 (3 x)}{2 (1)} + \frac{d}{d x} [1]$

Step 2.6.2.2

Cancel the common factor.

$\frac{2 (3 x)}{2 \cdot 1} + \frac{d}{d x} [1]$

Step 2.6.2.3

Rewrite the expression.

$\frac{3 x}{1} + \frac{d}{d x} [1]$

Step 2.6.2.4

Divide $3 x$ by $1$ .

$3 x + \frac{d}{d x} [1]$

Step 3

Differentiate using the Constant Rule.

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Step 3.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$3 x + 0$

Step 3.2

Add $3 x$ and $0$ .

$3 x$