Calculus Examples

$y = - 2 e^{2 x^{2} - 1}$

Step 1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 e^{2 x^{2} - 1}$ with respect to $x$ is $- 2 \frac{d}{d x} [e^{2 x^{2} - 1}]$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 2 x^{2} - 1$ .

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Step 2.1

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

$- 2 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [2 x^{2} - 1])$

Step 2.2

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

$- 2 (e^{u} \frac{d}{d x} [2 x^{2} - 1])$

Step 2.3

Replace all occurrences of $u$ with $2 x^{2} - 1$ .

$- 2 (e^{2 x^{2} - 1} \frac{d}{d x} [2 x^{2} - 1])$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $2 x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 1]$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Multiply $2$ by $2$ .

$- 2 e^{2 x^{2} - 1} (4 x + \frac{d}{d x} [- 1])$

Step 3.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$- 2 e^{2 x^{2} - 1} (4 x + 0)$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $4 x$ and $0$ .

$- 2 e^{2 x^{2} - 1} (4 x)$

Step 3.6.2

Multiply $4$ by $- 2$ .

$- 8 e^{2 x^{2} - 1} x$

Step 3.6.3

Reorder factors in $- 8 e^{2 x^{2} - 1} x$ .

$- 8 x e^{2 x^{2} - 1}$