Calculus Examples

$y = \frac{8 {(9 - 3 x)}^{5}}{5}$

Step 1

Since $\frac{8}{5}$ is constant with respect to $x$ , the derivative of $\frac{8 {(9 - 3 x)}^{5}}{5}$ with respect to $x$ is $\frac{8}{5} \frac{d}{d x} [{(9 - 3 x)}^{5}]$ .

$\frac{8}{5} \frac{d}{d x} [{(9 - 3 x)}^{5}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = 9 - 3 x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $9 - 3 x$ .

$\frac{8}{5} (\frac{d}{d u} [u^{5}] \frac{d}{d x} [9 - 3 x])$

Step 2.2

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

$\frac{8}{5} (5 u^{4} \frac{d}{d x} [9 - 3 x])$

Step 2.3

Replace all occurrences of $u$ with $9 - 3 x$ .

$\frac{8}{5} (5 {(9 - 3 x)}^{4} \frac{d}{d x} [9 - 3 x])$

Step 3

Differentiate.

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Step 3.1

Combine $5$ and $\frac{8}{5}$ .

$\frac{5 \cdot 8}{5} ({(9 - 3 x)}^{4} \frac{d}{d x} [9 - 3 x])$

Step 3.2

Simplify terms.

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Step 3.2.1

Multiply $5$ by $8$ .

$\frac{40}{5} ({(9 - 3 x)}^{4} \frac{d}{d x} [9 - 3 x])$

Step 3.2.2

Combine ${(9 - 3 x)}^{4}$ and $\frac{40}{5}$ .

$\frac{{(9 - 3 x)}^{4} \cdot 40}{5} \frac{d}{d x} [9 - 3 x]$

Step 3.2.3

Move $40$ to the left of ${(9 - 3 x)}^{4}$ .

$\frac{40 \cdot {(9 - 3 x)}^{4}}{5} \frac{d}{d x} [9 - 3 x]$

Step 3.2.4

Cancel the common factor of $40$ and $5$ .

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Step 3.2.4.1

Factor $5$ out of $40 {(9 - 3 x)}^{4}$ .

$\frac{5 (8 {(9 - 3 x)}^{4})}{5} \frac{d}{d x} [9 - 3 x]$

Step 3.2.4.2

Cancel the common factors.

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Step 3.2.4.2.1

Factor $5$ out of $5$ .

$\frac{5 (8 {(9 - 3 x)}^{4})}{5 (1)} \frac{d}{d x} [9 - 3 x]$

Step 3.2.4.2.2

Cancel the common factor.

$\frac{5 (8 {(9 - 3 x)}^{4})}{5 \cdot 1} \frac{d}{d x} [9 - 3 x]$

Step 3.2.4.2.3

Rewrite the expression.

$\frac{8 {(9 - 3 x)}^{4}}{1} \frac{d}{d x} [9 - 3 x]$

Step 3.2.4.2.4

Divide $8 {(9 - 3 x)}^{4}$ by $1$ .

$8 {(9 - 3 x)}^{4} \frac{d}{d x} [9 - 3 x]$

Step 3.3

By the Sum Rule, the derivative of $9 - 3 x$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- 3 x]$ .

$8 {(9 - 3 x)}^{4} (\frac{d}{d x} [9] + \frac{d}{d x} [- 3 x])$

Step 3.4

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$8 {(9 - 3 x)}^{4} (0 + \frac{d}{d x} [- 3 x])$

Step 3.5

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

$8 {(9 - 3 x)}^{4} \frac{d}{d x} [- 3 x]$

Step 3.6

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

$8 {(9 - 3 x)}^{4} (- 3 \frac{d}{d x} [x])$

Step 3.7

Multiply $- 3$ by $8$ .

$- 24 {(9 - 3 x)}^{4} \frac{d}{d x} [x]$

Step 3.8

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

$- 24 {(9 - 3 x)}^{4} \cdot 1$

Step 3.9

Multiply $- 24$ by $1$ .

$- 24 {(9 - 3 x)}^{4}$