Calculus Examples

$y = (1 + 2 x) e^{- 2 x}$

Step 1

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

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

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 $g (x) = - 2 x$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Simplify the expression.

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Step 3.3.1

Multiply $- 2$ by $1$ .

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

Step 3.3.2

Move $- 2$ to the left of $(1 + 2 x) e^{- 2 x}$ .

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

Step 3.4

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

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

Step 3.5

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

$- 2 (1 + 2 x) e^{- 2 x} + e^{- 2 x} (0 + \frac{d}{d x} [2 x])$

Step 3.6

Add $0$ and $\frac{d}{d x} [2 x]$ .

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

Simplify the expression.

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Step 3.9.1

Multiply $2$ by $1$ .

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

Step 3.9.2

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

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Combine terms.

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Step 4.3.1

Multiply $- 2$ by $1$ .

$- 2 e^{- 2 x} - 2 (2 x) e^{- 2 x} + 2 e^{- 2 x}$

Step 4.3.2

Multiply $2$ by $- 2$ .

$- 2 e^{- 2 x} - 4 x e^{- 2 x} + 2 e^{- 2 x}$

Step 4.3.3

Add $- 2 e^{- 2 x}$ and $2 e^{- 2 x}$ .

$- 4 x e^{- 2 x} + 0$

Step 4.3.4

Add $- 4 x e^{- 2 x}$ and $0$ .

$- 4 x e^{- 2 x}$

Step 4.4

Reorder the factors of $- 4 x e^{- 2 x}$ .

$- 4 e^{- 2 x} x$

Step 4.5

Reorder factors in $- 4 e^{- 2 x} x$ .

$- 4 x e^{- 2 x}$