Calculus Examples

$6 {(6 e^{8 x} + 5)}^{2}$

Step 1

Since $6$ is constant with respect to $x$ , the derivative of $6 {(6 e^{8 x} + 5)}^{2}$ with respect to $x$ is $6 \frac{d}{d x} [{(6 e^{8 x} + 5)}^{2}]$ .

$6 \frac{d}{d x} [{(6 e^{8 x} + 5)}^{2}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = 6 e^{8 x} + 5$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $6 e^{8 x} + 5$ .

$6 (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [6 e^{8 x} + 5])$

Step 2.2

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

$6 (2 u_{1} \frac{d}{d x} [6 e^{8 x} + 5])$

Step 2.3

Replace all occurrences of $u_{1}$ with $6 e^{8 x} + 5$ .

$6 (2 (6 e^{8 x} + 5) \frac{d}{d x} [6 e^{8 x} + 5])$

Step 3

Differentiate.

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Step 3.1

Multiply $2$ by $6$ .

$12 ((6 e^{8 x} + 5) \frac{d}{d x} [6 e^{8 x} + 5])$

Step 3.2

By the Sum Rule, the derivative of $6 e^{8 x} + 5$ with respect to $x$ is $\frac{d}{d x} [6 e^{8 x}] + \frac{d}{d x} [5]$ .

$12 (6 e^{8 x} + 5) (\frac{d}{d x} [6 e^{8 x}] + \frac{d}{d x} [5])$

Step 3.3

Since $6$ is constant with respect to $x$ , the derivative of $6 e^{8 x}$ with respect to $x$ is $6 \frac{d}{d x} [e^{8 x}]$ .

$12 (6 e^{8 x} + 5) (6 \frac{d}{d x} [e^{8 x}] + \frac{d}{d x} [5])$

Step 4

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 $g (x) = 8 x$ .

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Step 4.1

To apply the Chain Rule, set $u_{2}$ as $8 x$ .

$12 (6 e^{8 x} + 5) (6 (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d x} [8 x]) + \frac{d}{d x} [5])$

Step 4.2

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

$12 (6 e^{8 x} + 5) (6 (e^{u_{2}} \frac{d}{d x} [8 x]) + \frac{d}{d x} [5])$

Step 4.3

Replace all occurrences of $u_{2}$ with $8 x$ .

$12 (6 e^{8 x} + 5) (6 (e^{8 x} \frac{d}{d x} [8 x]) + \frac{d}{d x} [5])$

Step 5

Differentiate.

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Step 5.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$12 (6 e^{8 x} + 5) (6 e^{8 x} (8 \frac{d}{d x} [x]) + \frac{d}{d x} [5])$

Step 5.2

Multiply $8$ by $6$ .

$12 (6 e^{8 x} + 5) (48 e^{8 x} \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 5.3

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

$12 (6 e^{8 x} + 5) (48 e^{8 x} \cdot 1 + \frac{d}{d x} [5])$

Step 5.4

Multiply $48$ by $1$ .

$12 (6 e^{8 x} + 5) (48 e^{8 x} + \frac{d}{d x} [5])$

Step 5.5

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

$12 (6 e^{8 x} + 5) (48 e^{8 x} + 0)$

Step 5.6

Simplify the expression.

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Step 5.6.1

Add $48 e^{8 x}$ and $0$ .

$12 (6 e^{8 x} + 5) (48 e^{8 x})$

Step 5.6.2

Multiply $48$ by $12$ .

$576 (6 e^{8 x} + 5) e^{8 x}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$(576 (6 e^{8 x}) + 576 \cdot 5) e^{8 x}$

Step 6.2

Apply the distributive property.

$576 (6 e^{8 x}) e^{8 x} + 576 \cdot 5 e^{8 x}$

Step 6.3

Combine terms.

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Step 6.3.1

Multiply $6$ by $576$ .

$3456 e^{8 x} e^{8 x} + 576 \cdot 5 e^{8 x}$

Step 6.3.2

Multiply $e^{8 x}$ by $e^{8 x}$ by adding the exponents.

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Step 6.3.2.1

Move $e^{8 x}$ .

$3456 (e^{8 x} e^{8 x}) + 576 \cdot 5 e^{8 x}$

Step 6.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3456 e^{8 x + 8 x} + 576 \cdot 5 e^{8 x}$

Step 6.3.2.3

Add $8 x$ and $8 x$ .

$3456 e^{16 x} + 576 \cdot 5 e^{8 x}$

Step 6.3.3

Multiply $576$ by $5$ .

$3456 e^{16 x} + 2880 e^{8 x}$