Calculus Examples

$y = 8 x^{2} {(1 - 3 x)}^{5}$

Step 1

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{2} {(1 - 3 x)}^{5}$ with respect to $x$ is $8 \frac{d}{d x} [x^{2} {(1 - 3 x)}^{5}]$ .

$8 \frac{d}{d x} [x^{2} {(1 - 3 x)}^{5}]$

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 $g (x) = {(1 - 3 x)}^{5}$ .

$8 (x^{2} \frac{d}{d x} [{(1 - 3 x)}^{5}] + {(1 - 3 x)}^{5} \frac{d}{d x} [x^{2}])$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = 1 - 3 x$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $1 - 3 x$ .

$8 (x^{2} (\frac{d}{d u} [u^{5}] \frac{d}{d x} [1 - 3 x]) + {(1 - 3 x)}^{5} \frac{d}{d x} [x^{2}])$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 5$ .

$8 (x^{2} (5 u^{4} \frac{d}{d x} [1 - 3 x]) + {(1 - 3 x)}^{5} \frac{d}{d x} [x^{2}])$

Step 3.3

Replace all occurrences of $u$ with $1 - 3 x$ .

$8 (x^{2} (5 {(1 - 3 x)}^{4} \frac{d}{d x} [1 - 3 x]) + {(1 - 3 x)}^{5} \frac{d}{d x} [x^{2}])$

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $1 - 3 x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 3 x]$ .

$8 (x^{2} (5 {(1 - 3 x)}^{4} (\frac{d}{d x} [1] + \frac{d}{d x} [- 3 x])) + {(1 - 3 x)}^{5} \frac{d}{d x} [x^{2}])$

Step 4.2

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

$8 (x^{2} (5 {(1 - 3 x)}^{4} (0 + \frac{d}{d x} [- 3 x])) + {(1 - 3 x)}^{5} \frac{d}{d x} [x^{2}])$

Step 4.3

Add $0$ and $\frac{d}{d x} [- 3 x]$ .

$8 (x^{2} (5 {(1 - 3 x)}^{4} \frac{d}{d x} [- 3 x]) + {(1 - 3 x)}^{5} \frac{d}{d x} [x^{2}])$

Step 4.4

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

$8 (x^{2} (5 {(1 - 3 x)}^{4} (- 3 \frac{d}{d x} [x])) + {(1 - 3 x)}^{5} \frac{d}{d x} [x^{2}])$

Step 4.5

Multiply $- 3$ by $5$ .

$8 (x^{2} (- 15 {(1 - 3 x)}^{4} \frac{d}{d x} [x]) + {(1 - 3 x)}^{5} \frac{d}{d x} [x^{2}])$

Step 4.6

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

$8 (x^{2} (- 15 {(1 - 3 x)}^{4} \cdot 1) + {(1 - 3 x)}^{5} \frac{d}{d x} [x^{2}])$

Step 4.7

Multiply $- 15$ by $1$ .

$8 (x^{2} (- 15 {(1 - 3 x)}^{4}) + {(1 - 3 x)}^{5} \frac{d}{d x} [x^{2}])$

Step 4.8

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

$8 (x^{2} (- 15 {(1 - 3 x)}^{4}) + {(1 - 3 x)}^{5} (2 x))$

Step 4.9

Move $2$ to the left of ${(1 - 3 x)}^{5}$ .

$8 (x^{2} (- 15 {(1 - 3 x)}^{4}) + 2 \cdot {(1 - 3 x)}^{5} x)$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$8 (x^{2} (- 15 {(1 - 3 x)}^{4})) + 8 (2 {(1 - 3 x)}^{5} x)$

Step 5.2

Multiply $- 15$ by $8$ .

$- 120 (x^{2} ({(1 - 3 x)}^{4})) + 8 (2 {(1 - 3 x)}^{5} x)$

Step 5.3

Multiply $2$ by $8$ .

$- 120 x^{2} {(1 - 3 x)}^{4} + 16 ({(1 - 3 x)}^{5} x)$

Step 5.4

Factor $8 x {(1 - 3 x)}^{4}$ out of $- 120 x^{2} {(1 - 3 x)}^{4} + 16 {(1 - 3 x)}^{5} x$ .

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Step 5.4.1

Factor $8 x {(1 - 3 x)}^{4}$ out of $- 120 x^{2} {(1 - 3 x)}^{4}$ .

$8 x {(1 - 3 x)}^{4} (- 15 x) + 16 {(1 - 3 x)}^{5} x$

Step 5.4.2

Factor $8 x {(1 - 3 x)}^{4}$ out of $16 {(1 - 3 x)}^{5} x$ .

$8 x {(1 - 3 x)}^{4} (- 15 x) + 8 x {(1 - 3 x)}^{4} (2 (1 - 3 x))$

Step 5.4.3

Factor $8 x {(1 - 3 x)}^{4}$ out of $8 x {(1 - 3 x)}^{4} (- 15 x) + 8 x {(1 - 3 x)}^{4} (2 (1 - 3 x))$ .

$8 x {(1 - 3 x)}^{4} (- 15 x + 2 (1 - 3 x))$