Calculus Examples

$m (x) = 12 {(x^{2} + 5)}^{\frac{3}{2}} - 93$

Step 1

By the Sum Rule, the derivative of $12 {(x^{2} + 5)}^{\frac{3}{2}} - 93$ with respect to $x$ is $\frac{d}{d x} [12 {(x^{2} + 5)}^{\frac{3}{2}}] + \frac{d}{d x} [- 93]$ .

$\frac{d}{d x} [12 {(x^{2} + 5)}^{\frac{3}{2}}] + \frac{d}{d x} [- 93]$

Step 2

Evaluate $\frac{d}{d x} [12 {(x^{2} + 5)}^{\frac{3}{2}}]$ .

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Step 2.1

Since $12$ is constant with respect to $x$ , the derivative of $12 {(x^{2} + 5)}^{\frac{3}{2}}$ with respect to $x$ is $12 \frac{d}{d x} [{(x^{2} + 5)}^{\frac{3}{2}}]$ .

$12 \frac{d}{d x} [{(x^{2} + 5)}^{\frac{3}{2}}] + \frac{d}{d x} [- 93]$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{3}{2}}$ and $g (x) = x^{2} + 5$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $x^{2} + 5$ .

$12 (\frac{d}{d u} [u^{\frac{3}{2}}] \frac{d}{d x} [x^{2} + 5]) + \frac{d}{d x} [- 93]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{3}{2}$ .

$12 (\frac{3}{2} u^{\frac{3}{2} - 1} \frac{d}{d x} [x^{2} + 5]) + \frac{d}{d x} [- 93]$

Step 2.2.3

Replace all occurrences of $u$ with $x^{2} + 5$ .

$12 (\frac{3}{2} {(x^{2} + 5)}^{\frac{3}{2} - 1} \frac{d}{d x} [x^{2} + 5]) + \frac{d}{d x} [- 93]$

Step 2.3

By the Sum Rule, the derivative of $x^{2} + 5$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [5]$ .

$12 (\frac{3}{2} {(x^{2} + 5)}^{\frac{3}{2} - 1} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [5])) + \frac{d}{d x} [- 93]$

Step 2.4

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

$12 (\frac{3}{2} {(x^{2} + 5)}^{\frac{3}{2} - 1} (2 x + \frac{d}{d x} [5])) + \frac{d}{d x} [- 93]$

Step 2.5

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$12 (\frac{3}{2} {(x^{2} + 5)}^{\frac{3}{2} - 1} (2 x + 0)) + \frac{d}{d x} [- 93]$

Step 2.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$12 (\frac{3}{2} {(x^{2} + 5)}^{\frac{3}{2} - 1 \cdot \frac{2}{2}} (2 x + 0)) + \frac{d}{d x} [- 93]$

Step 2.7

Combine $- 1$ and $\frac{2}{2}$ .

$12 (\frac{3}{2} {(x^{2} + 5)}^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}} (2 x + 0)) + \frac{d}{d x} [- 93]$

Step 2.8

Combine the numerators over the common denominator.

$12 (\frac{3}{2} {(x^{2} + 5)}^{\frac{3 - 1 \cdot 2}{2}} (2 x + 0)) + \frac{d}{d x} [- 93]$

Step 2.9

Simplify the numerator.

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Step 2.9.1

Multiply $- 1$ by $2$ .

$12 (\frac{3}{2} {(x^{2} + 5)}^{\frac{3 - 2}{2}} (2 x + 0)) + \frac{d}{d x} [- 93]$

Step 2.9.2

Subtract $2$ from $3$ .

$12 (\frac{3}{2} {(x^{2} + 5)}^{\frac{1}{2}} (2 x + 0)) + \frac{d}{d x} [- 93]$

Step 2.10

Add $2 x$ and $0$ .

$12 (\frac{3}{2} {(x^{2} + 5)}^{\frac{1}{2}} (2 x)) + \frac{d}{d x} [- 93]$

Step 2.11

Combine $\frac{3}{2}$ and ${(x^{2} + 5)}^{\frac{1}{2}}$ .

$12 (\frac{3 {(x^{2} + 5)}^{\frac{1}{2}}}{2} (2 x)) + \frac{d}{d x} [- 93]$

Step 2.12

Combine $2$ and $\frac{3 {(x^{2} + 5)}^{\frac{1}{2}}}{2}$ .

$12 (\frac{2 (3 {(x^{2} + 5)}^{\frac{1}{2}})}{2} x) + \frac{d}{d x} [- 93]$

Step 2.13

Multiply $3$ by $2$ .

$12 (\frac{6 {(x^{2} + 5)}^{\frac{1}{2}}}{2} x) + \frac{d}{d x} [- 93]$

Step 2.14

Combine $\frac{6 {(x^{2} + 5)}^{\frac{1}{2}}}{2}$ and $x$ .

$12 \frac{6 {(x^{2} + 5)}^{\frac{1}{2}} x}{2} + \frac{d}{d x} [- 93]$

Step 2.15

Factor $2$ out of $6 {(x^{2} + 5)}^{\frac{1}{2}} x$ .

$12 \frac{2 (3 {(x^{2} + 5)}^{\frac{1}{2}} x)}{2} + \frac{d}{d x} [- 93]$

Step 2.16

Cancel the common factors.

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Step 2.16.1

Factor $2$ out of $2$ .

$12 \frac{2 (3 {(x^{2} + 5)}^{\frac{1}{2}} x)}{2 (1)} + \frac{d}{d x} [- 93]$

Step 2.16.2

Cancel the common factor.

$12 \frac{2 (3 {(x^{2} + 5)}^{\frac{1}{2}} x)}{2 \cdot 1} + \frac{d}{d x} [- 93]$

Step 2.16.3

Rewrite the expression.

$12 \frac{3 {(x^{2} + 5)}^{\frac{1}{2}} x}{1} + \frac{d}{d x} [- 93]$

Step 2.16.4

Divide $3 {(x^{2} + 5)}^{\frac{1}{2}} x$ by $1$ .

$12 (3 {(x^{2} + 5)}^{\frac{1}{2}} x) + \frac{d}{d x} [- 93]$

Step 2.17

Multiply $3$ by $12$ .

$36 {(x^{2} + 5)}^{\frac{1}{2}} x + \frac{d}{d x} [- 93]$

Step 3

Since $- 93$ is constant with respect to $x$ , the derivative of $- 93$ with respect to $x$ is $0$ .

$36 {(x^{2} + 5)}^{\frac{1}{2}} x + 0$

Step 4

Simplify.

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Step 4.1

Add $36 {(x^{2} + 5)}^{\frac{1}{2}} x$ and $0$ .

$36 {(x^{2} + 5)}^{\frac{1}{2}} x$

Step 4.2

Reorder the factors of $36 {(x^{2} + 5)}^{\frac{1}{2}} x$ .

$36 x {(x^{2} + 5)}^{\frac{1}{2}}$