Calculus Examples

$x^{2} {(x - 2)}^{4}$

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 $f (x) = x^{2}$ and $g (x) = {(x - 2)}^{4}$ .

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

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u$ as $x - 2$ .

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

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 4$ .

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

Step 2.3

Replace all occurrences of $u$ with $x - 2$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $1$ and $0$ .

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

Step 3.4.2

Multiply $4$ by $1$ .

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

Step 3.5

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

$x^{2} (4 {(x - 2)}^{3}) + {(x - 2)}^{4} (2 x)$

Step 3.6

Move $2$ to the left of ${(x - 2)}^{4}$ .

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

Step 4

Simplify.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.2

Combine terms.

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Step 4.2.1

Move $2$ to the left of $x$ .

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

Step 4.2.2

Move $2$ to the left of $x$ .

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

Step 4.2.3

Add $2 x$ and $x$ .

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