Calculus Examples

${(6 e^{2 x} - x)}^{3}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = 6 e^{2 x} - x$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $6 e^{2 x} - x$ .

$\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d x} [6 e^{2 x} - x]$

Step 1.2

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

$3 {u_{1}}^{2} \frac{d}{d x} [6 e^{2 x} - x]$

Step 1.3

Replace all occurrences of $u_{1}$ with $6 e^{2 x} - x$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $6 e^{2 x} - x$ with respect to $x$ is $\frac{d}{d x} [6 e^{2 x}] + \frac{d}{d x} [- x]$ .

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

Step 2.2

Since $6$ is constant with respect to $x$ , the derivative of $6 e^{2 x}$ with respect to $x$ is $6 \frac{d}{d x} [e^{2 x}]$ .

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

Step 3

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

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Step 3.1

To apply the Chain Rule, set $u_{2}$ as $2 x$ .

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

Step 3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

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

Step 3.3

Replace all occurrences of $u_{2}$ with $2 x$ .

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

Step 4

Differentiate.

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Step 4.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 4.2

Multiply $2$ by $6$ .

$3 {(6 e^{2 x} - x)}^{2} (12 e^{2 x} \frac{d}{d x} [x] + \frac{d}{d x} [- x])$

Step 4.3

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

$3 {(6 e^{2 x} - x)}^{2} (12 e^{2 x} \cdot 1 + \frac{d}{d x} [- x])$

Step 4.4

Multiply $12$ by $1$ .

$3 {(6 e^{2 x} - x)}^{2} (12 e^{2 x} + \frac{d}{d x} [- x])$

Step 4.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$3 {(6 e^{2 x} - x)}^{2} (12 e^{2 x} - \frac{d}{d x} [x])$

Step 4.6

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

$3 {(6 e^{2 x} - x)}^{2} (12 e^{2 x} - 1 \cdot 1)$

Step 4.7

Multiply $- 1$ by $1$ .

$3 {(6 e^{2 x} - x)}^{2} (12 e^{2 x} - 1)$