Calculus Examples

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

Step 1

Differentiate.

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Step 1.1

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

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

Step 1.2

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

$0 + \frac{d}{d x} [- e^{2 x}]$

Step 2

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

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Step 2.1

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

$0 - \frac{d}{d x} [e^{2 x}]$

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 $f (x) = e^{x}$ and $g (x) = 2 x$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $2 x$ .

$0 - (\frac{d}{d u} [e^{u}] \frac{d}{d x} [2 x])$

Step 2.2.2

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

$0 - (e^{u} \frac{d}{d x} [2 x])$

Step 2.2.3

Replace all occurrences of $u$ with $2 x$ .

$0 - (e^{2 x} \frac{d}{d x} [2 x])$

Step 2.3

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

$0 - (e^{2 x} (2 \frac{d}{d x} [x]))$

Step 2.4

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

$0 - (e^{2 x} (2 \cdot 1))$

Step 2.5

Multiply $2$ by $1$ .

$0 - (e^{2 x} \cdot 2)$

Step 2.6

Move $2$ to the left of $e^{2 x}$ .

$0 - (2 \cdot e^{2 x})$

Step 2.7

Multiply $2$ by $- 1$ .

$0 - 2 e^{2 x}$

Step 3

Subtract $2 e^{2 x}$ from $0$ .

$- 2 e^{2 x}$