Calculus Examples

$y (x) = e^{x^{3} - 8}$ , $x = 2$

Step 1

Find the derivative.

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Step 1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x^{3} - 8$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $x^{3} - 8$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [x^{3} - 8]$

Step 1.1.2

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

$e^{u} \frac{d}{d x} [x^{3} - 8]$

Step 1.1.3

Replace all occurrences of $u$ with $x^{3} - 8$ .

$e^{x^{3} - 8} \frac{d}{d x} [x^{3} - 8]$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $x^{3} - 8$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 8]$ .

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

Step 1.2.2

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

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

Step 1.2.3

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$e^{x^{3} - 8} (3 x^{2} + 0)$

Step 1.2.4

Add $3 x^{2}$ and $0$ .

$e^{x^{3} - 8} (3 x^{2})$

Step 1.3

Simplify.

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Step 1.3.1

Reorder the factors of $e^{x^{3} - 8} (3 x^{2})$ .

$3 e^{x^{3} - 8} x^{2}$

Step 1.3.2

Reorder factors in $3 e^{x^{3} - 8} x^{2}$ .

$3 x^{2} e^{x^{3} - 8}$

Step 2

Replace the variable $x$ with $2$ in the expression.

$y' (2) = 3 {(2)}^{2} e^{{(2)}^{3} - 8}$

Step 3

Raise $2$ to the power of $2$ .

$y' (2) = 3 \cdot (4 e^{{(2)}^{3} - 8})$

Step 4

Multiply $3$ by $4$ .

$y' (2) = 12 e^{{(2)}^{3} - 8}$

Step 5

Raise $2$ to the power of $3$ .

$y' (2) = 12 e^{8 - 8}$

Step 6

Subtract $8$ from $8$ .

$y' (2) = 12 e^{0}$

Step 7

Anything raised to $0$ is $1$ .

$y' (2) = 12 \cdot 1$

Step 8

Multiply $12$ by $1$ .

$y' (2) = 12$