Calculus Examples

$y = \frac{1}{4} \cdot (x e^{4 x}) - \frac{1}{16} \cdot e^{4 x}$

Step 1

By the Sum Rule, the derivative of $\frac{1}{4} \cdot (x e^{4 x}) - \frac{1}{16} \cdot e^{4 x}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{4} \cdot (x e^{4 x})] + \frac{d}{d x} [- \frac{1}{16} \cdot e^{4 x}]$ .

$\frac{d}{d x} [\frac{1}{4} \cdot (x e^{4 x})] + \frac{d}{d x} [- \frac{1}{16} \cdot e^{4 x}]$

Step 2

Evaluate $\frac{d}{d x} [\frac{1}{4} \cdot (x e^{4 x})]$ .

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Step 2.1

Combine $x$ and $\frac{1}{4}$ .

$\frac{d}{d x} [\frac{x}{4} \cdot e^{4 x}] + \frac{d}{d x} [- \frac{1}{16} \cdot e^{4 x}]$

Step 2.2

Combine $\frac{x}{4}$ and $e^{4 x}$ .

$\frac{d}{d x} [\frac{x e^{4 x}}{4}] + \frac{d}{d x} [- \frac{1}{16} \cdot e^{4 x}]$

Step 2.3

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{x e^{4 x}}{4}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [x e^{4 x}]$ .

$\frac{1}{4} \frac{d}{d x} [x e^{4 x}] + \frac{d}{d x} [- \frac{1}{16} \cdot e^{4 x}]$

Step 2.4

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 $g (x) = e^{4 x}$ .

$\frac{1}{4} (x \frac{d}{d x} [e^{4 x}] + e^{4 x} \frac{d}{d x} [x]) + \frac{d}{d x} [- \frac{1}{16} \cdot e^{4 x}]$

Step 2.5

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 $g (x) = 4 x$ .

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Step 2.5.1

To apply the Chain Rule, set $u_{1}$ as $4 x$ .

$\frac{1}{4} (x (\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [4 x]) + e^{4 x} \frac{d}{d x} [x]) + \frac{d}{d x} [- \frac{1}{16} \cdot e^{4 x}]$

Step 2.5.2

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

$\frac{1}{4} (x (e^{u_{1}} \frac{d}{d x} [4 x]) + e^{4 x} \frac{d}{d x} [x]) + \frac{d}{d x} [- \frac{1}{16} \cdot e^{4 x}]$

Step 2.5.3

Replace all occurrences of $u_{1}$ with $4 x$ .

$\frac{1}{4} (x (e^{4 x} \frac{d}{d x} [4 x]) + e^{4 x} \frac{d}{d x} [x]) + \frac{d}{d x} [- \frac{1}{16} \cdot e^{4 x}]$

Step 2.6

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

$\frac{1}{4} (x (e^{4 x} (4 \frac{d}{d x} [x])) + e^{4 x} \frac{d}{d x} [x]) + \frac{d}{d x} [- \frac{1}{16} \cdot e^{4 x}]$

Step 2.7

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

$\frac{1}{4} (x (e^{4 x} (4 \cdot 1)) + e^{4 x} \frac{d}{d x} [x]) + \frac{d}{d x} [- \frac{1}{16} \cdot e^{4 x}]$

Step 2.8

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

$\frac{1}{4} (x (e^{4 x} (4 \cdot 1)) + e^{4 x} \cdot 1) + \frac{d}{d x} [- \frac{1}{16} \cdot e^{4 x}]$

Step 2.9

Multiply $4$ by $1$ .

$\frac{1}{4} (x (e^{4 x} \cdot 4) + e^{4 x} \cdot 1) + \frac{d}{d x} [- \frac{1}{16} \cdot e^{4 x}]$

Step 2.10

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

$\frac{1}{4} (x (4 \cdot e^{4 x}) + e^{4 x} \cdot 1) + \frac{d}{d x} [- \frac{1}{16} \cdot e^{4 x}]$

Step 2.11

Multiply $e^{4 x}$ by $1$ .

$\frac{1}{4} (x (4 e^{4 x}) + e^{4 x}) + \frac{d}{d x} [- \frac{1}{16} \cdot e^{4 x}]$

Step 3

Evaluate $\frac{d}{d x} [- \frac{1}{16} \cdot e^{4 x}]$ .

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Step 3.1

Since $- \frac{1}{16}$ is constant with respect to $x$ , the derivative of $- \frac{1}{16} \cdot e^{4 x}$ with respect to $x$ is $- \frac{1}{16} \frac{d}{d x} [e^{4 x}]$ .

$\frac{1}{4} (x (4 e^{4 x}) + e^{4 x}) - \frac{1}{16} \frac{d}{d x} [e^{4 x}]$

Step 3.2

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 $g (x) = 4 x$ .

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Step 3.2.1

To apply the Chain Rule, set $u_{2}$ as $4 x$ .

$\frac{1}{4} (x (4 e^{4 x}) + e^{4 x}) - \frac{1}{16} (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d x} [4 x])$

Step 3.2.2

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

$\frac{1}{4} (x (4 e^{4 x}) + e^{4 x}) - \frac{1}{16} (e^{u_{2}} \frac{d}{d x} [4 x])$

Step 3.2.3

Replace all occurrences of $u_{2}$ with $4 x$ .

$\frac{1}{4} (x (4 e^{4 x}) + e^{4 x}) - \frac{1}{16} (e^{4 x} \frac{d}{d x} [4 x])$

Step 3.3

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

$\frac{1}{4} (x (4 e^{4 x}) + e^{4 x}) - \frac{1}{16} (e^{4 x} (4 \frac{d}{d x} [x]))$

Step 3.4

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

$\frac{1}{4} (x (4 e^{4 x}) + e^{4 x}) - \frac{1}{16} (e^{4 x} (4 \cdot 1))$

Step 3.5

Multiply $4$ by $1$ .

$\frac{1}{4} (x (4 e^{4 x}) + e^{4 x}) - \frac{1}{16} (e^{4 x} \cdot 4)$

Step 3.6

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

$\frac{1}{4} (x (4 e^{4 x}) + e^{4 x}) - \frac{1}{16} (4 \cdot e^{4 x})$

Step 3.7

Multiply $4$ by $- 1$ .

$\frac{1}{4} (x (4 e^{4 x}) + e^{4 x}) - 4 (\frac{1}{16}) e^{4 x}$

Step 3.8

Combine $- 4$ and $\frac{1}{16}$ .

$\frac{1}{4} (x (4 e^{4 x}) + e^{4 x}) + \frac{- 4}{16} e^{4 x}$

Step 3.9

Combine $\frac{- 4}{16}$ and $e^{4 x}$ .

$\frac{1}{4} (x (4 e^{4 x}) + e^{4 x}) + \frac{- 4 e^{4 x}}{16}$

Step 3.10

Cancel the common factor of $- 4$ and $16$ .

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Step 3.10.1

Factor $4$ out of $- 4 e^{4 x}$ .

$\frac{1}{4} (x (4 e^{4 x}) + e^{4 x}) + \frac{4 (- e^{4 x})}{16}$

Step 3.10.2

Cancel the common factors.

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Step 3.10.2.1

Factor $4$ out of $16$ .

$\frac{1}{4} (x (4 e^{4 x}) + e^{4 x}) + \frac{4 (- e^{4 x})}{4 (4)}$

Step 3.10.2.2

Cancel the common factor.

$\frac{1}{4} (x (4 e^{4 x}) + e^{4 x}) + \frac{4 (- e^{4 x})}{4 \cdot 4}$

Step 3.10.2.3

Rewrite the expression.

$\frac{1}{4} (x (4 e^{4 x}) + e^{4 x}) + \frac{- e^{4 x}}{4}$

Step 3.11

Move the negative in front of the fraction.

$\frac{1}{4} (x (4 e^{4 x}) + e^{4 x}) - \frac{e^{4 x}}{4}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{1}{4} (x (4 e^{4 x})) + \frac{1}{4} e^{4 x} - \frac{e^{4 x}}{4}$

Step 4.2

Combine terms.

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Step 4.2.1

Combine $x$ and $\frac{1}{4}$ .

$\frac{x}{4} (4 e^{4 x}) + \frac{1}{4} e^{4 x} - \frac{e^{4 x}}{4}$

Step 4.2.2

Combine $4$ and $\frac{x}{4}$ .

$\frac{4 x}{4} e^{4 x} + \frac{1}{4} e^{4 x} - \frac{e^{4 x}}{4}$

Step 4.2.3

Combine $\frac{4 x}{4}$ and $e^{4 x}$ .

$\frac{4 x e^{4 x}}{4} + \frac{1}{4} e^{4 x} - \frac{e^{4 x}}{4}$

Step 4.2.4

Cancel the common factor of $4$ .

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Step 4.2.4.1

Cancel the common factor.

$\frac{4 x e^{4 x}}{4} + \frac{1}{4} e^{4 x} - \frac{e^{4 x}}{4}$

Step 4.2.4.2

Divide $x e^{4 x}$ by $1$ .

$x e^{4 x} + \frac{1}{4} e^{4 x} - \frac{e^{4 x}}{4}$

Step 4.2.5

To write $x e^{4 x}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{1}{4} e^{4 x} + x e^{4 x} \cdot \frac{4}{4} - \frac{e^{4 x}}{4}$

Step 4.2.6

Combine $x e^{4 x}$ and $\frac{4}{4}$ .

$\frac{1}{4} e^{4 x} + \frac{x e^{4 x} \cdot 4}{4} - \frac{e^{4 x}}{4}$

Step 4.2.7

Combine the numerators over the common denominator.

$\frac{1}{4} e^{4 x} + \frac{x e^{4 x} \cdot 4 - e^{4 x}}{4}$

Step 4.2.8

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

$\frac{1}{4} e^{4 x} + \frac{4 \cdot (x e^{4 x}) - e^{4 x}}{4}$

Step 4.2.9

Reorder terms.

$\frac{1}{4} e^{4 x} + \frac{4 e^{4 x} x - e^{4 x}}{4}$

Step 4.2.10

To write $\frac{1}{4} e^{4 x}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{1}{4} e^{4 x} \cdot \frac{4}{4} + \frac{4 e^{4 x} x - e^{4 x}}{4}$

Step 4.2.11

Combine $\frac{1}{4} e^{4 x}$ and $\frac{4}{4}$ .

$\frac{\frac{1}{4} e^{4 x} \cdot 4}{4} + \frac{4 e^{4 x} x - e^{4 x}}{4}$

Step 4.2.12

Combine the numerators over the common denominator.

$\frac{\frac{1}{4} e^{4 x} \cdot 4 + 4 e^{4 x} x - e^{4 x}}{4}$

Step 4.2.13

Combine $\frac{1}{4}$ and $e^{4 x}$ .

$\frac{\frac{e^{4 x}}{4} \cdot 4 + 4 e^{4 x} x - e^{4 x}}{4}$

Step 4.2.14

Combine $\frac{e^{4 x}}{4}$ and $4$ .

$\frac{\frac{e^{4 x} \cdot 4}{4} + 4 e^{4 x} x - e^{4 x}}{4}$

Step 4.2.15

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

$\frac{\frac{4 \cdot e^{4 x}}{4} + 4 e^{4 x} x - e^{4 x}}{4}$

Step 4.2.16

Cancel the common factor of $4$ .

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Step 4.2.16.1

Cancel the common factor.

$\frac{\frac{4 e^{4 x}}{4} + 4 e^{4 x} x - e^{4 x}}{4}$

Step 4.2.16.2

Divide $e^{4 x}$ by $1$ .

$\frac{e^{4 x} + 4 e^{4 x} x - e^{4 x}}{4}$

Step 4.2.17

Subtract $e^{4 x}$ from $e^{4 x}$ .

$\frac{4 e^{4 x} x + 0}{4}$

Step 4.2.18

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

$\frac{4 e^{4 x} x}{4}$

Step 4.2.19

Cancel the common factor of $4$ .

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Step 4.2.19.1

Cancel the common factor.

$\frac{4 e^{4 x} x}{4}$

Step 4.2.19.2

Divide $e^{4 x} x$ by $1$ .

$e^{4 x} x$

Step 4.3

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

$x e^{4 x}$