Calculus Examples

$f (x) = 3 x^{2} {(6 - 9 x)}^{4}$

Step 1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2} {(6 - 9 x)}^{4}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2} {(6 - 9 x)}^{4}]$ .

$3 \frac{d}{d x} [x^{2} {(6 - 9 x)}^{4}]$

Step 2

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 $g (x) = {(6 - 9 x)}^{4}$ .

$3 (x^{2} \frac{d}{d x} [{(6 - 9 x)}^{4}] + {(6 - 9 x)}^{4} \frac{d}{d x} [x^{2}])$

Step 3

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 $g (x) = 6 - 9 x$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $6 - 9 x$ .

$3 (x^{2} (\frac{d}{d u} [u^{4}] \frac{d}{d x} [6 - 9 x]) + {(6 - 9 x)}^{4} \frac{d}{d x} [x^{2}])$

Step 3.2

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

$3 (x^{2} (4 u^{3} \frac{d}{d x} [6 - 9 x]) + {(6 - 9 x)}^{4} \frac{d}{d x} [x^{2}])$

Step 3.3

Replace all occurrences of $u$ with $6 - 9 x$ .

$3 (x^{2} (4 {(6 - 9 x)}^{3} \frac{d}{d x} [6 - 9 x]) + {(6 - 9 x)}^{4} \frac{d}{d x} [x^{2}])$

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $6 - 9 x$ with respect to $x$ is $\frac{d}{d x} [6] + \frac{d}{d x} [- 9 x]$ .

$3 (x^{2} (4 {(6 - 9 x)}^{3} (\frac{d}{d x} [6] + \frac{d}{d x} [- 9 x])) + {(6 - 9 x)}^{4} \frac{d}{d x} [x^{2}])$

Step 4.2

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$3 (x^{2} (4 {(6 - 9 x)}^{3} (0 + \frac{d}{d x} [- 9 x])) + {(6 - 9 x)}^{4} \frac{d}{d x} [x^{2}])$

Step 4.3

Add $0$ and $\frac{d}{d x} [- 9 x]$ .

$3 (x^{2} (4 {(6 - 9 x)}^{3} \frac{d}{d x} [- 9 x]) + {(6 - 9 x)}^{4} \frac{d}{d x} [x^{2}])$

Step 4.4

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x$ with respect to $x$ is $- 9 \frac{d}{d x} [x]$ .

$3 (x^{2} (4 {(6 - 9 x)}^{3} (- 9 \frac{d}{d x} [x])) + {(6 - 9 x)}^{4} \frac{d}{d x} [x^{2}])$

Step 4.5

Multiply $- 9$ by $4$ .

$3 (x^{2} (- 36 {(6 - 9 x)}^{3} \frac{d}{d x} [x]) + {(6 - 9 x)}^{4} \frac{d}{d x} [x^{2}])$

Step 4.6

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

$3 (x^{2} (- 36 {(6 - 9 x)}^{3} \cdot 1) + {(6 - 9 x)}^{4} \frac{d}{d x} [x^{2}])$

Step 4.7

Multiply $- 36$ by $1$ .

$3 (x^{2} (- 36 {(6 - 9 x)}^{3}) + {(6 - 9 x)}^{4} \frac{d}{d x} [x^{2}])$

Step 4.8

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

$3 (x^{2} (- 36 {(6 - 9 x)}^{3}) + {(6 - 9 x)}^{4} (2 x))$

Step 4.9

Move $2$ to the left of ${(6 - 9 x)}^{4}$ .

$3 (x^{2} (- 36 {(6 - 9 x)}^{3}) + 2 \cdot {(6 - 9 x)}^{4} x)$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$3 (x^{2} (- 36 {(6 - 9 x)}^{3})) + 3 (2 {(6 - 9 x)}^{4} x)$

Step 5.2

Multiply $- 36$ by $3$ .

$- 108 (x^{2} ({(6 - 9 x)}^{3})) + 3 (2 {(6 - 9 x)}^{4} x)$

Step 5.3

Multiply $2$ by $3$ .

$- 108 x^{2} {(6 - 9 x)}^{3} + 6 ({(6 - 9 x)}^{4} x)$

Step 5.4

Factor $6 x {(6 - 9 x)}^{3}$ out of $- 108 x^{2} {(6 - 9 x)}^{3} + 6 {(6 - 9 x)}^{4} x$ .

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Step 5.4.1

Factor $6 x {(6 - 9 x)}^{3}$ out of $- 108 x^{2} {(6 - 9 x)}^{3}$ .

$6 x {(6 - 9 x)}^{3} (- 18 x) + 6 {(6 - 9 x)}^{4} x$

Step 5.4.2

Factor $6 x {(6 - 9 x)}^{3}$ out of $6 {(6 - 9 x)}^{4} x$ .

$6 x {(6 - 9 x)}^{3} (- 18 x) + 6 x {(6 - 9 x)}^{3} (6 - 9 x)$

Step 5.4.3

Factor $6 x {(6 - 9 x)}^{3}$ out of $6 x {(6 - 9 x)}^{3} (- 18 x) + 6 x {(6 - 9 x)}^{3} (6 - 9 x)$ .

$6 x {(6 - 9 x)}^{3} (- 18 x + 6 - 9 x)$

Step 5.5

Subtract $9 x$ from $- 18 x$ .

$6 x {(6 - 9 x)}^{3} (- 27 x + 6)$