Calculus Examples

$\arctan (4 x^{2} - 3)$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \arctan (x)$ and $g (x) = 4 x^{2} - 3$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $4 x^{2} - 3$ .

$\frac{d}{d u} [\arctan (u)] \frac{d}{d x} [4 x^{2} - 3]$

Step 1.2

The derivative of $\arctan (u)$ with respect to $u$ is $\frac{1}{1 + u^{2}}$ .

$\frac{1}{1 + u^{2}} \frac{d}{d x} [4 x^{2} - 3]$

Step 1.3

Replace all occurrences of $u$ with $4 x^{2} - 3$ .

$\frac{1}{1 + {(4 x^{2} - 3)}^{2}} \frac{d}{d x} [4 x^{2} - 3]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $4 x^{2} - 3$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 3]$ .

$\frac{1}{1 + {(4 x^{2} - 3)}^{2}} (\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 3])$

Step 2.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$\frac{1}{1 + {(4 x^{2} - 3)}^{2}} (4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3])$

Step 2.3

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

$\frac{1}{1 + {(4 x^{2} - 3)}^{2}} (4 (2 x) + \frac{d}{d x} [- 3])$

Step 2.4

Multiply $2$ by $4$ .

$\frac{1}{1 + {(4 x^{2} - 3)}^{2}} (8 x + \frac{d}{d x} [- 3])$

Step 2.5

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

$\frac{1}{1 + {(4 x^{2} - 3)}^{2}} (8 x + 0)$

Step 2.6

Combine fractions.

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Step 2.6.1

Add $8 x$ and $0$ .

$\frac{1}{1 + {(4 x^{2} - 3)}^{2}} (8 x)$

Step 2.6.2

Combine $8$ and $\frac{1}{1 + {(4 x^{2} - 3)}^{2}}$ .

$\frac{8}{1 + {(4 x^{2} - 3)}^{2}} x$

Step 2.6.3

Combine $\frac{8}{1 + {(4 x^{2} - 3)}^{2}}$ and $x$ .

$\frac{8 x}{1 + {(4 x^{2} - 3)}^{2}}$

Step 3

Simplify.

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Step 3.1

Reorder terms.

$\frac{8 x}{{(4 x^{2} - 3)}^{2} + 1}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Rewrite ${(4 x^{2} - 3)}^{2}$ as $(4 x^{2} - 3) (4 x^{2} - 3)$ .

$\frac{8 x}{(4 x^{2} - 3) (4 x^{2} - 3) + 1}$

Step 3.2.2

Expand $(4 x^{2} - 3) (4 x^{2} - 3)$ using the FOIL Method.

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Step 3.2.2.1

Apply the distributive property.

$\frac{8 x}{4 x^{2} (4 x^{2} - 3) - 3 (4 x^{2} - 3) + 1}$

Step 3.2.2.2

Apply the distributive property.

$\frac{8 x}{4 x^{2} (4 x^{2}) + 4 x^{2} \cdot - 3 - 3 (4 x^{2} - 3) + 1}$

Step 3.2.2.3

Apply the distributive property.

$\frac{8 x}{4 x^{2} (4 x^{2}) + 4 x^{2} \cdot - 3 - 3 (4 x^{2}) - 3 \cdot - 3 + 1}$

Step 3.2.3

Simplify and combine like terms.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{8 x}{4 \cdot 4 x^{2} x^{2} + 4 x^{2} \cdot - 3 - 3 (4 x^{2}) - 3 \cdot - 3 + 1}$

Step 3.2.3.1.2

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 3.2.3.1.2.1

Move $x^{2}$ .

$\frac{8 x}{4 \cdot 4 (x^{2} x^{2}) + 4 x^{2} \cdot - 3 - 3 (4 x^{2}) - 3 \cdot - 3 + 1}$

Step 3.2.3.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{8 x}{4 \cdot 4 x^{2 + 2} + 4 x^{2} \cdot - 3 - 3 (4 x^{2}) - 3 \cdot - 3 + 1}$

Step 3.2.3.1.2.3

Add $2$ and $2$ .

$\frac{8 x}{4 \cdot 4 x^{4} + 4 x^{2} \cdot - 3 - 3 (4 x^{2}) - 3 \cdot - 3 + 1}$

Step 3.2.3.1.3

Multiply $4$ by $4$ .

$\frac{8 x}{16 x^{4} + 4 x^{2} \cdot - 3 - 3 (4 x^{2}) - 3 \cdot - 3 + 1}$

Step 3.2.3.1.4

Multiply $- 3$ by $4$ .

$\frac{8 x}{16 x^{4} - 12 x^{2} - 3 (4 x^{2}) - 3 \cdot - 3 + 1}$

Step 3.2.3.1.5

Multiply $4$ by $- 3$ .

$\frac{8 x}{16 x^{4} - 12 x^{2} - 12 x^{2} - 3 \cdot - 3 + 1}$

Step 3.2.3.1.6

Multiply $- 3$ by $- 3$ .

$\frac{8 x}{16 x^{4} - 12 x^{2} - 12 x^{2} + 9 + 1}$

Step 3.2.3.2

Subtract $12 x^{2}$ from $- 12 x^{2}$ .

$\frac{8 x}{16 x^{4} - 24 x^{2} + 9 + 1}$

Step 3.2.4

Add $9$ and $1$ .

$\frac{8 x}{16 x^{4} - 24 x^{2} + 10}$

Step 3.2.5

Factor $2$ out of $16 x^{4} - 24 x^{2} + 10$ .

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Step 3.2.5.1

Factor $2$ out of $16 x^{4}$ .

$\frac{8 x}{2 (8 x^{4}) - 24 x^{2} + 10}$

Step 3.2.5.2

Factor $2$ out of $- 24 x^{2}$ .

$\frac{8 x}{2 (8 x^{4}) + 2 (- 12 x^{2}) + 10}$

Step 3.2.5.3

Factor $2$ out of $10$ .

$\frac{8 x}{2 (8 x^{4}) + 2 (- 12 x^{2}) + 2 (5)}$

Step 3.2.5.4

Factor $2$ out of $2 (8 x^{4}) + 2 (- 12 x^{2})$ .

$\frac{8 x}{2 (8 x^{4} - 12 x^{2}) + 2 (5)}$

Step 3.2.5.5

Factor $2$ out of $2 (8 x^{4} - 12 x^{2}) + 2 (5)$ .

$\frac{8 x}{2 (8 x^{4} - 12 x^{2} + 5)}$

Step 3.3

Cancel the common factor of $8$ and $2$ .

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Step 3.3.1

Factor $2$ out of $8 x$ .

$\frac{2 (4 x)}{2 (8 x^{4} - 12 x^{2} + 5)}$

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Cancel the common factor.

$\frac{2 (4 x)}{2 (8 x^{4} - 12 x^{2} + 5)}$

Step 3.3.2.2

Rewrite the expression.

$\frac{4 x}{8 x^{4} - 12 x^{2} + 5}$