Calculus Examples

${(x^{2} + 1)}^{2}$

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{2} + 1$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $x^{2} + 1$ .

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

Step 1.1.2

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

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

Step 1.1.3

Replace all occurrences of $u$ with $x^{2} + 1$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

$2 (x^{2} + 1) (2 x + 0)$

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Add $2 x$ and $0$ .

$2 (x^{2} + 1) (2 x)$

Step 1.2.4.2

Multiply $2$ by $2$ .

$4 (x^{2} + 1) x$

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

$(4 x^{2} + 4 \cdot 1) x$

Step 1.3.2

Apply the distributive property.

$4 x^{2} x + 4 \cdot 1 x$

Step 1.3.3

Combine terms.

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Step 1.3.3.1

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

$4 (x^{1} x^{2}) + 4 \cdot 1 x$

Step 1.3.3.2

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

$4 x^{1 + 2} + 4 \cdot 1 x$

Step 1.3.3.3

Add $1$ and $2$ .

$4 x^{3} + 4 \cdot 1 x$

Step 1.3.3.4

Multiply $4$ by $1$ .

$f' (x) = 4 x^{3} + 4 x$

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

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Step 2.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $3$ by $4$ .

$12 x^{2} + \frac{d}{d x} [4 x]$

Step 2.3

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 2.3.1

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

$12 x^{2} + 4 \frac{d}{d x} [x]$

Step 2.3.2

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

$12 x^{2} + 4 \cdot 1$

Step 2.3.3

Multiply $4$ by $1$ .

$f'' (x) = 12 x^{2} + 4$