Calculus Examples

$g (x) = {(x^{2} + 1)}^{3} {(x^{2} + 2)}^{6}$

Step 1

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 $f (x) = {(x^{2} + 1)}^{3}$ and $g (x) = {(x^{2} + 2)}^{6}$ .

${(x^{2} + 1)}^{3} \frac{d}{d x} [{(x^{2} + 2)}^{6}] + {(x^{2} + 2)}^{6} \frac{d}{d x} [{(x^{2} + 1)}^{3}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{6}$ and $g (x) = x^{2} + 2$ .

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Step 2.1

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

${(x^{2} + 1)}^{3} (\frac{d}{d u_{1}} [{u_{1}}^{6}] \frac{d}{d x} [x^{2} + 2]) + {(x^{2} + 2)}^{6} \frac{d}{d x} [{(x^{2} + 1)}^{3}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 6$ .

${(x^{2} + 1)}^{3} (6 {u_{1}}^{5} \frac{d}{d x} [x^{2} + 2]) + {(x^{2} + 2)}^{6} \frac{d}{d x} [{(x^{2} + 1)}^{3}]$

Step 2.3

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

${(x^{2} + 1)}^{3} (6 {(x^{2} + 2)}^{5} \frac{d}{d x} [x^{2} + 2]) + {(x^{2} + 2)}^{6} \frac{d}{d x} [{(x^{2} + 1)}^{3}]$

Step 3

Differentiate.

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Step 3.1

Move $6$ to the left of ${(x^{2} + 1)}^{3}$ .

$6 \cdot {(x^{2} + 1)}^{3} {(x^{2} + 2)}^{5} \frac{d}{d x} [x^{2} + 2] + {(x^{2} + 2)}^{6} \frac{d}{d x} [{(x^{2} + 1)}^{3}]$

Step 3.2

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

$6 {(x^{2} + 1)}^{3} {(x^{2} + 2)}^{5} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2]) + {(x^{2} + 2)}^{6} \frac{d}{d x} [{(x^{2} + 1)}^{3}]$

Step 3.3

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

$6 {(x^{2} + 1)}^{3} {(x^{2} + 2)}^{5} (2 x + \frac{d}{d x} [2]) + {(x^{2} + 2)}^{6} \frac{d}{d x} [{(x^{2} + 1)}^{3}]$

Step 3.4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$6 {(x^{2} + 1)}^{3} {(x^{2} + 2)}^{5} (2 x + 0) + {(x^{2} + 2)}^{6} \frac{d}{d x} [{(x^{2} + 1)}^{3}]$

Step 3.5

Simplify the expression.

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Step 3.5.1

Add $2 x$ and $0$ .

$6 {(x^{2} + 1)}^{3} {(x^{2} + 2)}^{5} (2 x) + {(x^{2} + 2)}^{6} \frac{d}{d x} [{(x^{2} + 1)}^{3}]$

Step 3.5.2

Multiply $2$ by $6$ .

$12 {(x^{2} + 1)}^{3} {(x^{2} + 2)}^{5} x + {(x^{2} + 2)}^{6} \frac{d}{d x} [{(x^{2} + 1)}^{3}]$

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = x^{2} + 1$ .

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Step 4.1

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

$12 {(x^{2} + 1)}^{3} {(x^{2} + 2)}^{5} x + {(x^{2} + 2)}^{6} (\frac{d}{d u_{2}} [{u_{2}}^{3}] \frac{d}{d x} [x^{2} + 1])$

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 3$ .

$12 {(x^{2} + 1)}^{3} {(x^{2} + 2)}^{5} x + {(x^{2} + 2)}^{6} (3 {u_{2}}^{2} \frac{d}{d x} [x^{2} + 1])$

Step 4.3

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

$12 {(x^{2} + 1)}^{3} {(x^{2} + 2)}^{5} x + {(x^{2} + 2)}^{6} (3 {(x^{2} + 1)}^{2} \frac{d}{d x} [x^{2} + 1])$

Step 5

Differentiate.

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Step 5.1

Move $3$ to the left of ${(x^{2} + 2)}^{6}$ .

$12 {(x^{2} + 1)}^{3} {(x^{2} + 2)}^{5} x + 3 \cdot {(x^{2} + 2)}^{6} {(x^{2} + 1)}^{2} \frac{d}{d x} [x^{2} + 1]$

Step 5.2

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

$12 {(x^{2} + 1)}^{3} {(x^{2} + 2)}^{5} x + 3 {(x^{2} + 2)}^{6} {(x^{2} + 1)}^{2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1])$

Step 5.3

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

$12 {(x^{2} + 1)}^{3} {(x^{2} + 2)}^{5} x + 3 {(x^{2} + 2)}^{6} {(x^{2} + 1)}^{2} (2 x + \frac{d}{d x} [1])$

Step 5.4

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

$12 {(x^{2} + 1)}^{3} {(x^{2} + 2)}^{5} x + 3 {(x^{2} + 2)}^{6} {(x^{2} + 1)}^{2} (2 x + 0)$

Step 5.5

Simplify the expression.

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Step 5.5.1

Add $2 x$ and $0$ .

$12 {(x^{2} + 1)}^{3} {(x^{2} + 2)}^{5} x + 3 {(x^{2} + 2)}^{6} {(x^{2} + 1)}^{2} (2 x)$

Step 5.5.2

Multiply $2$ by $3$ .

$12 {(x^{2} + 1)}^{3} {(x^{2} + 2)}^{5} x + 6 {(x^{2} + 2)}^{6} {(x^{2} + 1)}^{2} x$

Step 6

Simplify.

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Step 6.1

Factor $6 {(x^{2} + 1)}^{2} {(x^{2} + 2)}^{5} x$ out of $12 {(x^{2} + 1)}^{3} {(x^{2} + 2)}^{5} x + 6 {(x^{2} + 2)}^{6} {(x^{2} + 1)}^{2} x$ .

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Step 6.1.1

Factor $6 {(x^{2} + 1)}^{2} {(x^{2} + 2)}^{5} x$ out of $12 {(x^{2} + 1)}^{3} {(x^{2} + 2)}^{5} x$ .

$6 {(x^{2} + 1)}^{2} {(x^{2} + 2)}^{5} x (2 (x^{2} + 1)) + 6 {(x^{2} + 2)}^{6} {(x^{2} + 1)}^{2} x$

Step 6.1.2

Factor $6 {(x^{2} + 1)}^{2} {(x^{2} + 2)}^{5} x$ out of $6 {(x^{2} + 2)}^{6} {(x^{2} + 1)}^{2} x$ .

$6 {(x^{2} + 1)}^{2} {(x^{2} + 2)}^{5} x (2 (x^{2} + 1)) + 6 {(x^{2} + 1)}^{2} {(x^{2} + 2)}^{5} x (x^{2} + 2)$

Step 6.1.3

Factor $6 {(x^{2} + 1)}^{2} {(x^{2} + 2)}^{5} x$ out of $6 {(x^{2} + 1)}^{2} {(x^{2} + 2)}^{5} x (2 (x^{2} + 1)) + 6 {(x^{2} + 1)}^{2} {(x^{2} + 2)}^{5} x (x^{2} + 2)$ .

$6 {(x^{2} + 1)}^{2} {(x^{2} + 2)}^{5} x (2 (x^{2} + 1) + x^{2} + 2)$

Step 6.2

Reorder the factors of $6 {(x^{2} + 1)}^{2} {(x^{2} + 2)}^{5} x (2 (x^{2} + 1) + x^{2} + 2)$ .

$6 x {(x^{2} + 2)}^{5} {(x^{2} + 1)}^{2} (2 (x^{2} + 1) + x^{2} + 2)$