Calculus Examples

$f (x) = (4 x - 2) e^{2 x}$

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 $f (x) = 4 x - 2$ and $g (x) = e^{2 x}$ .

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

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 $2 x$ .

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

Step 2.2

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

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

Step 2.3

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

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

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 $2 \frac{d}{d x} [x]$ .

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

Step 3.2

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

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

Step 3.3

Simplify the expression.

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Step 3.3.1

Multiply $2$ by $1$ .

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

Step 3.3.2

Move $2$ to the left of $(4 x - 2) e^{2 x}$ .

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

Step 3.4

By the Sum Rule, the derivative of $4 x - 2$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 2]$ .

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

Step 3.5

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

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

Step 3.6

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

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

Step 3.7

Multiply $4$ by $1$ .

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

Step 3.8

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

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

Step 3.9

Simplify the expression.

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Step 3.9.1

Add $4$ and $0$ .

$2 (4 x - 2) e^{2 x} + e^{2 x} \cdot 4$

Step 3.9.2

Move $4$ to the left of $e^{2 x}$ .

$2 (4 x - 2) e^{2 x} + 4 \cdot e^{2 x}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$(2 (4 x) + 2 \cdot - 2) e^{2 x} + 4 e^{2 x}$

Step 4.2

Apply the distributive property.

$2 (4 x) e^{2 x} + 2 \cdot - 2 e^{2 x} + 4 e^{2 x}$

Step 4.3

Combine terms.

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Step 4.3.1

Multiply $4$ by $2$ .

$8 x e^{2 x} + 2 \cdot - 2 e^{2 x} + 4 e^{2 x}$

Step 4.3.2

Multiply $2$ by $- 2$ .

$8 x e^{2 x} - 4 e^{2 x} + 4 e^{2 x}$

Step 4.3.3

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

$8 x e^{2 x} + 0$

Step 4.3.4

Add $8 x e^{2 x}$ and $0$ .

$8 x e^{2 x}$

Step 4.4

Reorder the factors of $8 x e^{2 x}$ .

$8 e^{2 x} x$

Step 4.5

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

$8 x e^{2 x}$