Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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 $g (x) = - x$ .

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Step 2.3.1

To apply the Chain Rule, set $u_{1}$ as $- x$ .

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

Step 2.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

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

Step 2.3.3

Replace all occurrences of $u_{1}$ with $- x$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Multiply $- 1$ by $1$ .

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

Step 2.8

Move $- 1$ to the left of $e^{- x}$ .

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

Step 2.9

Rewrite $- 1 e^{- x}$ as $- e^{- x}$ .

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

Step 2.10

Multiply $e^{- x}$ by $1$ .

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

Step 3

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

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Step 3.1

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 $g (x) = 2 x$ .

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Multiply $2$ by $1$ .

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

Step 3.5

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

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $- 1$ by $2$ .

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

Step 4.3

Reorder terms.

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

Step 4.4

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

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