Calculus Examples

$\frac{4 x^{3} + 3 x^{2}}{x}$

Step 1

Simplify terms.

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Step 1.1

Factor $x^{2}$ out of $4 x^{3} + 3 x^{2}$ .

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Step 1.1.1

Factor $x^{2}$ out of $4 x^{3}$ .

$\frac{d}{d x} [\frac{x^{2} (4 x) + 3 x^{2}}{x}]$

Step 1.1.2

Factor $x^{2}$ out of $3 x^{2}$ .

$\frac{d}{d x} [\frac{x^{2} (4 x) + x^{2} \cdot 3}{x}]$

Step 1.1.3

Factor $x^{2}$ out of $x^{2} (4 x) + x^{2} \cdot 3$ .

$\frac{d}{d x} [\frac{x^{2} (4 x + 3)}{x}]$

Step 1.2

Cancel the common factor of $x^{2}$ and $x$ .

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Step 1.2.1

Factor $x$ out of $x^{2} (4 x + 3)$ .

$\frac{d}{d x} [\frac{x (x (4 x + 3))}{x}]$

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Raise $x$ to the power of $1$ .

$\frac{d}{d x} [\frac{x (x (4 x + 3))}{x^{1}}]$

Step 1.2.2.2

Factor $x$ out of $x^{1}$ .

$\frac{d}{d x} [\frac{x (x (4 x + 3))}{x \cdot 1}]$

Step 1.2.2.3

Cancel the common factor.

$\frac{d}{d x} [\frac{x (x (4 x + 3))}{x \cdot 1}]$

Step 1.2.2.4

Rewrite the expression.

$\frac{d}{d x} [\frac{x (4 x + 3)}{1}]$

Step 1.2.2.5

Divide $x (4 x + 3)$ by $1$ .

$\frac{d}{d x} [x (4 x + 3)]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = 4 x + 3$ .

$x \frac{d}{d x} [4 x + 3] + (4 x + 3) \frac{d}{d x} [x]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $4 x + 3$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [3]$ .

$x (\frac{d}{d x} [4 x] + \frac{d}{d x} [3]) + (4 x + 3) \frac{d}{d x} [x]$

Step 3.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$x (4 \frac{d}{d x} [x] + \frac{d}{d x} [3]) + (4 x + 3) \frac{d}{d x} [x]$

Step 3.3

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

$x (4 \cdot 1 + \frac{d}{d x} [3]) + (4 x + 3) \frac{d}{d x} [x]$

Step 3.4

Multiply $4$ by $1$ .

$x (4 + \frac{d}{d x} [3]) + (4 x + 3) \frac{d}{d x} [x]$

Step 3.5

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$x (4 + 0) + (4 x + 3) \frac{d}{d x} [x]$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $4$ and $0$ .

$x \cdot 4 + (4 x + 3) \frac{d}{d x} [x]$

Step 3.6.2

Move $4$ to the left of $x$ .

$4 \cdot x + (4 x + 3) \frac{d}{d x} [x]$

Step 3.7

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

$4 x + (4 x + 3) \cdot 1$

Step 3.8

Simplify by adding terms.

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Step 3.8.1

Multiply $4 x + 3$ by $1$ .

$4 x + 4 x + 3$

Step 3.8.2

Add $4 x$ and $4 x$ .

$8 x + 3$