Calculus Examples

$(x - 1) {(x^{2} + 2)}^{3}$

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Differentiate.

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Step 3.1

Move $3$ to the left of $x - 1$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Simplify the expression.

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Step 3.5.1

Add $2 x$ and $0$ .

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

Step 3.5.2

Multiply $2$ by $3$ .

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

Simplify the expression.

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Step 3.9.1

Add $1$ and $0$ .

$6 (x - 1) {(x^{2} + 2)}^{2} x + {(x^{2} + 2)}^{3} \cdot 1$

Step 3.9.2

Multiply ${(x^{2} + 2)}^{3}$ by $1$ .

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$(6 x + 6 \cdot - 1) {(x^{2} + 2)}^{2} x + {(x^{2} + 2)}^{3}$

Step 4.2

Multiply $6$ by $- 1$ .

$(6 x - 6) {(x^{2} + 2)}^{2} x + {(x^{2} + 2)}^{3}$

Step 4.3

Factor ${(x^{2} + 2)}^{2}$ out of $(6 x - 6) {(x^{2} + 2)}^{2} x + {(x^{2} + 2)}^{3}$ .

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Step 4.3.1

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

${(x^{2} + 2)}^{2} ((6 x - 6) x) + {(x^{2} + 2)}^{3}$

Step 4.3.2

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

${(x^{2} + 2)}^{2} ((6 x - 6) x) + {(x^{2} + 2)}^{2} (x^{2} + 2)$

Step 4.3.3

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

${(x^{2} + 2)}^{2} ((6 x - 6) x + x^{2} + 2)$