Calculus Examples

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

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 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 $f (x) = x^{\frac{1}{2}} + 1$ and $g (x) = e^{- 2 x}$ .

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

Step 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 $g (x) = - 2 x$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Simplify the expression.

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Step 4.3.1

Multiply $- 2$ by $1$ .

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

Step 4.3.2

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

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

Step 4.4

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

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

Step 4.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

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

Step 5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 6

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $2$ .

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

Step 8.2

Subtract $2$ from $1$ .

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

Step 9

Combine fractions.

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Step 9.1

Move the negative in front of the fraction.

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

Step 9.2

Combine $\frac{1}{2}$ and $x^{- \frac{1}{2}}$ .

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

Step 9.3

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 10

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

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

Step 11

Combine fractions.

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Step 11.1

Add $\frac{1}{2 x^{\frac{1}{2}}}$ and $0$ .

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

Step 11.2

Combine $e^{- 2 x}$ and $\frac{1}{2 x^{\frac{1}{2}}}$ .

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

Step 12

To write $- 2 (x^{\frac{1}{2}} + 1) e^{- 2 x}$ as a fraction with a common denominator, multiply by $\frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$ .

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

Step 13

Combine $- 2 (x^{\frac{1}{2}} + 1) e^{- 2 x}$ and $\frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$ .

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

Step 14

Combine the numerators over the common denominator.

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

Step 15

Multiply $2$ by $- 2$ .

$\frac{- 4 (x^{\frac{1}{2}} + 1) e^{- 2 x} x^{\frac{1}{2}} + e^{- 2 x}}{2 x^{\frac{1}{2}}}$

Step 16

Simplify.

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Step 16.1

Apply the distributive property.

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

Step 16.2

Apply the distributive property.

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

Step 16.3

Apply the distributive property.

$\frac{- 4 x^{\frac{1}{2}} e^{- 2 x} x^{\frac{1}{2}} - 4 \cdot 1 e^{- 2 x} x^{\frac{1}{2}} + e^{- 2 x}}{2 x^{\frac{1}{2}}}$

Step 16.4

Simplify each term.

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Step 16.4.1

Multiply $x^{\frac{1}{2}}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 16.4.1.1

Move $x^{\frac{1}{2}}$ .

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

Step 16.4.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{- 4 x^{\frac{1}{2} + \frac{1}{2}} e^{- 2 x} - 4 \cdot 1 e^{- 2 x} x^{\frac{1}{2}} + e^{- 2 x}}{2 x^{\frac{1}{2}}}$

Step 16.4.1.3

Combine the numerators over the common denominator.

$\frac{- 4 x^{\frac{1 + 1}{2}} e^{- 2 x} - 4 \cdot 1 e^{- 2 x} x^{\frac{1}{2}} + e^{- 2 x}}{2 x^{\frac{1}{2}}}$

Step 16.4.1.4

Add $1$ and $1$ .

$\frac{- 4 x^{\frac{2}{2}} e^{- 2 x} - 4 \cdot 1 e^{- 2 x} x^{\frac{1}{2}} + e^{- 2 x}}{2 x^{\frac{1}{2}}}$

Step 16.4.1.5

Divide $2$ by $2$ .

$\frac{- 4 x^{1} e^{- 2 x} - 4 \cdot 1 e^{- 2 x} x^{\frac{1}{2}} + e^{- 2 x}}{2 x^{\frac{1}{2}}}$

Step 16.4.2

Simplify $- 4 x^{1} e^{- 2 x}$ .

$\frac{- 4 x e^{- 2 x} - 4 \cdot 1 e^{- 2 x} x^{\frac{1}{2}} + e^{- 2 x}}{2 x^{\frac{1}{2}}}$

Step 16.4.3

Multiply $- 4$ by $1$ .

$\frac{- 4 x e^{- 2 x} - 4 e^{- 2 x} x^{\frac{1}{2}} + e^{- 2 x}}{2 x^{\frac{1}{2}}}$

Step 16.5

Reorder terms.

$\frac{- 4 e^{- 2 x} x - 4 e^{- 2 x} x^{\frac{1}{2}} + e^{- 2 x}}{2 x^{\frac{1}{2}}}$

Step 16.6

Factor $e^{- 2 x}$ out of $- 4 e^{- 2 x} x - 4 e^{- 2 x} x^{\frac{1}{2}} + e^{- 2 x}$ .

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Step 16.6.1

Factor $e^{- 2 x}$ out of $- 4 e^{- 2 x} x$ .

$\frac{e^{- 2 x} (- 4 x) - 4 e^{- 2 x} x^{\frac{1}{2}} + e^{- 2 x}}{2 x^{\frac{1}{2}}}$

Step 16.6.2

Factor $e^{- 2 x}$ out of $- 4 e^{- 2 x} x^{\frac{1}{2}}$ .

$\frac{e^{- 2 x} (- 4 x) + e^{- 2 x} (- 4 x^{\frac{1}{2}}) + e^{- 2 x}}{2 x^{\frac{1}{2}}}$

Step 16.6.3

Multiply by $1$ .

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

Step 16.6.4

Factor $e^{- 2 x}$ out of $e^{- 2 x} (- 4 x) + e^{- 2 x} (- 4 x^{\frac{1}{2}})$ .

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

Step 16.6.5

Factor $e^{- 2 x}$ out of $e^{- 2 x} (- 4 x - 4 x^{\frac{1}{2}}) + e^{- 2 x} \cdot 1$ .

$\frac{e^{- 2 x} (- 4 x - 4 x^{\frac{1}{2}} + 1)}{2 x^{\frac{1}{2}}}$

Step 16.7

Factor $- 1$ out of $- 4 x$ .

$\frac{e^{- 2 x} (- (4 x) - 4 x^{\frac{1}{2}} + 1)}{2 x^{\frac{1}{2}}}$

Step 16.8

Factor $- 1$ out of $- 4 x^{\frac{1}{2}}$ .

$\frac{e^{- 2 x} (- (4 x) - (4 x^{\frac{1}{2}}) + 1)}{2 x^{\frac{1}{2}}}$

Step 16.9

Factor $- 1$ out of $- (4 x) - (4 x^{\frac{1}{2}})$ .

$\frac{e^{- 2 x} (- (4 x + 4 x^{\frac{1}{2}}) + 1)}{2 x^{\frac{1}{2}}}$

Step 16.10

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{e^{- 2 x} (- (4 x + 4 x^{\frac{1}{2}}) - 1 (- 1))}{2 x^{\frac{1}{2}}}$

Step 16.11

Factor $- 1$ out of $- (4 x + 4 x^{\frac{1}{2}}) - 1 (- 1)$ .

$\frac{e^{- 2 x} (- (4 x + 4 x^{\frac{1}{2}} - 1))}{2 x^{\frac{1}{2}}}$

Step 16.12

Rewrite $- (4 x + 4 x^{\frac{1}{2}} - 1)$ as $- 1 (4 x + 4 x^{\frac{1}{2}} - 1)$ .

$\frac{e^{- 2 x} (- 1 (4 x + 4 x^{\frac{1}{2}} - 1))}{2 x^{\frac{1}{2}}}$

Step 16.13

Move the negative in front of the fraction.

$- \frac{e^{- 2 x} (4 x + 4 x^{\frac{1}{2}} - 1)}{2 x^{\frac{1}{2}}}$