Calculus Examples

$e^{x \sqrt{7 x - 15}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 x - 15}$ as ${(7 x - 15)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [e^{x {(7 x - 15)}^{\frac{1}{2}}}]$

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 $g (x) = x {(7 x - 15)}^{\frac{1}{2}}$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $x {(7 x - 15)}^{\frac{1}{2}}$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [x {(7 x - 15)}^{\frac{1}{2}}]$

Step 2.2

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

$e^{u_{1}} \frac{d}{d x} [x {(7 x - 15)}^{\frac{1}{2}}]$

Step 2.3

Replace all occurrences of $u_{1}$ with $x {(7 x - 15)}^{\frac{1}{2}}$ .

$e^{x {(7 x - 15)}^{\frac{1}{2}}} \frac{d}{d x} [x {(7 x - 15)}^{\frac{1}{2}}]$

Step 3

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 $g (x) = {(7 x - 15)}^{\frac{1}{2}}$ .

$e^{x {(7 x - 15)}^{\frac{1}{2}}} (x \frac{d}{d x} [{(7 x - 15)}^{\frac{1}{2}}] + {(7 x - 15)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 7 x - 15$ .

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Step 4.1

To apply the Chain Rule, set $u_{2}$ as $7 x - 15$ .

$e^{x {(7 x - 15)}^{\frac{1}{2}}} (x (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{2}}] \frac{d}{d x} [7 x - 15]) + {(7 x - 15)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.2

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

$e^{x {(7 x - 15)}^{\frac{1}{2}}} (x (\frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d x} [7 x - 15]) + {(7 x - 15)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4.3

Replace all occurrences of $u_{2}$ with $7 x - 15$ .

$e^{x {(7 x - 15)}^{\frac{1}{2}}} (x (\frac{1}{2} {(7 x - 15)}^{\frac{1}{2} - 1} \frac{d}{d x} [7 x - 15]) + {(7 x - 15)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 5

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

$e^{x {(7 x - 15)}^{\frac{1}{2}}} (x (\frac{1}{2} {(7 x - 15)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [7 x - 15]) + {(7 x - 15)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 6

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

$e^{x {(7 x - 15)}^{\frac{1}{2}}} (x (\frac{1}{2} {(7 x - 15)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [7 x - 15]) + {(7 x - 15)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 7

Combine the numerators over the common denominator.

$e^{x {(7 x - 15)}^{\frac{1}{2}}} (x (\frac{1}{2} {(7 x - 15)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [7 x - 15]) + {(7 x - 15)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $2$ .

$e^{x {(7 x - 15)}^{\frac{1}{2}}} (x (\frac{1}{2} {(7 x - 15)}^{\frac{1 - 2}{2}} \frac{d}{d x} [7 x - 15]) + {(7 x - 15)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 8.2

Subtract $2$ from $1$ .

$e^{x {(7 x - 15)}^{\frac{1}{2}}} (x (\frac{1}{2} {(7 x - 15)}^{\frac{- 1}{2}} \frac{d}{d x} [7 x - 15]) + {(7 x - 15)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 9

Combine fractions.

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Step 9.1

Move the negative in front of the fraction.

$e^{x {(7 x - 15)}^{\frac{1}{2}}} (x (\frac{1}{2} {(7 x - 15)}^{- \frac{1}{2}} \frac{d}{d x} [7 x - 15]) + {(7 x - 15)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 9.2

Combine $\frac{1}{2}$ and ${(7 x - 15)}^{- \frac{1}{2}}$ .

$e^{x {(7 x - 15)}^{\frac{1}{2}}} (x (\frac{{(7 x - 15)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [7 x - 15]) + {(7 x - 15)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 9.3

Move ${(7 x - 15)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$e^{x {(7 x - 15)}^{\frac{1}{2}}} (x (\frac{1}{2 {(7 x - 15)}^{\frac{1}{2}}} \frac{d}{d x} [7 x - 15]) + {(7 x - 15)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 9.4

Combine $\frac{1}{2 {(7 x - 15)}^{\frac{1}{2}}}$ and $x$ .

$e^{x {(7 x - 15)}^{\frac{1}{2}}} (\frac{x}{2 {(7 x - 15)}^{\frac{1}{2}}} \frac{d}{d x} [7 x - 15] + {(7 x - 15)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 10

By the Sum Rule, the derivative of $7 x - 15$ with respect to $x$ is $\frac{d}{d x} [7 x] + \frac{d}{d x} [- 15]$ .

$e^{x {(7 x - 15)}^{\frac{1}{2}}} (\frac{x}{2 {(7 x - 15)}^{\frac{1}{2}}} (\frac{d}{d x} [7 x] + \frac{d}{d x} [- 15]) + {(7 x - 15)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 11

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$e^{x {(7 x - 15)}^{\frac{1}{2}}} (\frac{x}{2 {(7 x - 15)}^{\frac{1}{2}}} (7 \frac{d}{d x} [x] + \frac{d}{d x} [- 15]) + {(7 x - 15)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 12

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

$e^{x {(7 x - 15)}^{\frac{1}{2}}} (\frac{x}{2 {(7 x - 15)}^{\frac{1}{2}}} (7 \cdot 1 + \frac{d}{d x} [- 15]) + {(7 x - 15)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 13

Multiply $7$ by $1$ .

$e^{x {(7 x - 15)}^{\frac{1}{2}}} (\frac{x}{2 {(7 x - 15)}^{\frac{1}{2}}} (7 + \frac{d}{d x} [- 15]) + {(7 x - 15)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 14

Since $- 15$ is constant with respect to $x$ , the derivative of $- 15$ with respect to $x$ is $0$ .

$e^{x {(7 x - 15)}^{\frac{1}{2}}} (\frac{x}{2 {(7 x - 15)}^{\frac{1}{2}}} (7 + 0) + {(7 x - 15)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 15

Combine fractions.

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Step 15.1

Add $7$ and $0$ .

$e^{x {(7 x - 15)}^{\frac{1}{2}}} (\frac{x}{2 {(7 x - 15)}^{\frac{1}{2}}} \cdot 7 + {(7 x - 15)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 15.2

Combine $\frac{x}{2 {(7 x - 15)}^{\frac{1}{2}}}$ and $7$ .

$e^{x {(7 x - 15)}^{\frac{1}{2}}} (\frac{x \cdot 7}{2 {(7 x - 15)}^{\frac{1}{2}}} + {(7 x - 15)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 15.3

Move $7$ to the left of $x$ .

$e^{x {(7 x - 15)}^{\frac{1}{2}}} (\frac{7 \cdot x}{2 {(7 x - 15)}^{\frac{1}{2}}} + {(7 x - 15)}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 16

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

$e^{x {(7 x - 15)}^{\frac{1}{2}}} (\frac{7 x}{2 {(7 x - 15)}^{\frac{1}{2}}} + {(7 x - 15)}^{\frac{1}{2}} \cdot 1)$

Step 17

Multiply ${(7 x - 15)}^{\frac{1}{2}}$ by $1$ .

$e^{x {(7 x - 15)}^{\frac{1}{2}}} (\frac{7 x}{2 {(7 x - 15)}^{\frac{1}{2}}} + {(7 x - 15)}^{\frac{1}{2}})$

Step 18

To write ${(7 x - 15)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 {(7 x - 15)}^{\frac{1}{2}}}{2 {(7 x - 15)}^{\frac{1}{2}}}$ .

$e^{x {(7 x - 15)}^{\frac{1}{2}}} (\frac{7 x}{2 {(7 x - 15)}^{\frac{1}{2}}} + {(7 x - 15)}^{\frac{1}{2}} \cdot \frac{2 {(7 x - 15)}^{\frac{1}{2}}}{2 {(7 x - 15)}^{\frac{1}{2}}})$

Step 19

Combine ${(7 x - 15)}^{\frac{1}{2}}$ and $\frac{2 {(7 x - 15)}^{\frac{1}{2}}}{2 {(7 x - 15)}^{\frac{1}{2}}}$ .

$e^{x {(7 x - 15)}^{\frac{1}{2}}} (\frac{7 x}{2 {(7 x - 15)}^{\frac{1}{2}}} + \frac{{(7 x - 15)}^{\frac{1}{2}} (2 {(7 x - 15)}^{\frac{1}{2}})}{2 {(7 x - 15)}^{\frac{1}{2}}})$

Step 20

Combine the numerators over the common denominator.

$e^{x {(7 x - 15)}^{\frac{1}{2}}} \frac{7 x + {(7 x - 15)}^{\frac{1}{2}} (2 {(7 x - 15)}^{\frac{1}{2}})}{2 {(7 x - 15)}^{\frac{1}{2}}}$

Step 21

Multiply ${(7 x - 15)}^{\frac{1}{2}}$ by ${(7 x - 15)}^{\frac{1}{2}}$ by adding the exponents.

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Step 21.1

Move ${(7 x - 15)}^{\frac{1}{2}}$ .

$e^{x {(7 x - 15)}^{\frac{1}{2}}} \frac{7 x + {(7 x - 15)}^{\frac{1}{2}} {(7 x - 15)}^{\frac{1}{2}} \cdot 2}{2 {(7 x - 15)}^{\frac{1}{2}}}$

Step 21.2

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

$e^{x {(7 x - 15)}^{\frac{1}{2}}} \frac{7 x + {(7 x - 15)}^{\frac{1}{2} + \frac{1}{2}} \cdot 2}{2 {(7 x - 15)}^{\frac{1}{2}}}$

Step 21.3

Combine the numerators over the common denominator.

$e^{x {(7 x - 15)}^{\frac{1}{2}}} \frac{7 x + {(7 x - 15)}^{\frac{1 + 1}{2}} \cdot 2}{2 {(7 x - 15)}^{\frac{1}{2}}}$

Step 21.4

Add $1$ and $1$ .

$e^{x {(7 x - 15)}^{\frac{1}{2}}} \frac{7 x + {(7 x - 15)}^{\frac{2}{2}} \cdot 2}{2 {(7 x - 15)}^{\frac{1}{2}}}$

Step 21.5

Divide $2$ by $2$ .

$e^{x {(7 x - 15)}^{\frac{1}{2}}} \frac{7 x + {(7 x - 15)}^{1} \cdot 2}{2 {(7 x - 15)}^{\frac{1}{2}}}$

Step 22

Simplify the expression.

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Step 22.1

Simplify ${(7 x - 15)}^{1} \cdot 2$ .

$e^{x {(7 x - 15)}^{\frac{1}{2}}} \frac{7 x + (7 x - 15) \cdot 2}{2 {(7 x - 15)}^{\frac{1}{2}}}$

Step 22.2

Move $2$ to the left of $7 x - 15$ .

$e^{x {(7 x - 15)}^{\frac{1}{2}}} \frac{7 x + 2 \cdot (7 x - 15)}{2 {(7 x - 15)}^{\frac{1}{2}}}$

Step 23

Combine $e^{x {(7 x - 15)}^{\frac{1}{2}}}$ and $\frac{7 x + 2 (7 x - 15)}{2 {(7 x - 15)}^{\frac{1}{2}}}$ .

$\frac{e^{x {(7 x - 15)}^{\frac{1}{2}}} (7 x + 2 (7 x - 15))}{2 {(7 x - 15)}^{\frac{1}{2}}}$

Step 24

Simplify.

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Step 24.1

Apply the distributive property.

$\frac{e^{x {(7 x - 15)}^{\frac{1}{2}}} (7 x + 2 (7 x) + 2 \cdot - 15)}{2 {(7 x - 15)}^{\frac{1}{2}}}$

Step 24.2

Simplify the numerator.

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Step 24.2.1

Simplify each term.

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Step 24.2.1.1

Multiply $7$ by $2$ .

$\frac{e^{x {(7 x - 15)}^{\frac{1}{2}}} (7 x + 14 x + 2 \cdot - 15)}{2 {(7 x - 15)}^{\frac{1}{2}}}$

Step 24.2.1.2

Multiply $2$ by $- 15$ .

$\frac{e^{x {(7 x - 15)}^{\frac{1}{2}}} (7 x + 14 x - 30)}{2 {(7 x - 15)}^{\frac{1}{2}}}$

Step 24.2.2

Add $7 x$ and $14 x$ .

$\frac{e^{x {(7 x - 15)}^{\frac{1}{2}}} (21 x - 30)}{2 {(7 x - 15)}^{\frac{1}{2}}}$

Step 24.2.3

Apply the distributive property.

$\frac{e^{x {(7 x - 15)}^{\frac{1}{2}}} (21 x) + e^{x {(7 x - 15)}^{\frac{1}{2}}} \cdot - 30}{2 {(7 x - 15)}^{\frac{1}{2}}}$

Step 24.2.4

Rewrite using the commutative property of multiplication.

$\frac{21 e^{x {(7 x - 15)}^{\frac{1}{2}}} x + e^{x {(7 x - 15)}^{\frac{1}{2}}} \cdot - 30}{2 {(7 x - 15)}^{\frac{1}{2}}}$

Step 24.2.5

Move $- 30$ to the left of $e^{x {(7 x - 15)}^{\frac{1}{2}}}$ .

$\frac{21 e^{x {(7 x - 15)}^{\frac{1}{2}}} x - 30 \cdot e^{x {(7 x - 15)}^{\frac{1}{2}}}}{2 {(7 x - 15)}^{\frac{1}{2}}}$

Step 24.2.6

Reorder factors in $21 e^{x {(7 x - 15)}^{\frac{1}{2}}} x - 30 e^{x {(7 x - 15)}^{\frac{1}{2}}}$ .

$\frac{21 x e^{x {(7 x - 15)}^{\frac{1}{2}}} - 30 e^{x {(7 x - 15)}^{\frac{1}{2}}}}{2 {(7 x - 15)}^{\frac{1}{2}}}$

Step 24.3

Reorder terms.

$\frac{21 e^{x {(7 x - 15)}^{\frac{1}{2}}} x - 30 e^{x {(7 x - 15)}^{\frac{1}{2}}}}{2 {(7 x - 15)}^{\frac{1}{2}}}$

Step 24.4

Factor $3 e^{x {(7 x - 15)}^{\frac{1}{2}}}$ out of $21 e^{x {(7 x - 15)}^{\frac{1}{2}}} x - 30 e^{x {(7 x - 15)}^{\frac{1}{2}}}$ .

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Step 24.4.1

Move $e^{x {(7 x - 15)}^{\frac{1}{2}}}$ .

$\frac{21 x e^{x {(7 x - 15)}^{\frac{1}{2}}} - 30 e^{x {(7 x - 15)}^{\frac{1}{2}}}}{2 {(7 x - 15)}^{\frac{1}{2}}}$

Step 24.4.2

Factor $3 e^{x {(7 x - 15)}^{\frac{1}{2}}}$ out of $21 x e^{x {(7 x - 15)}^{\frac{1}{2}}}$ .

$\frac{3 e^{x {(7 x - 15)}^{\frac{1}{2}}} (7 x) - 30 e^{x {(7 x - 15)}^{\frac{1}{2}}}}{2 {(7 x - 15)}^{\frac{1}{2}}}$

Step 24.4.3

Factor $3 e^{x {(7 x - 15)}^{\frac{1}{2}}}$ out of $- 30 e^{x {(7 x - 15)}^{\frac{1}{2}}}$ .

$\frac{3 e^{x {(7 x - 15)}^{\frac{1}{2}}} (7 x) + 3 e^{x {(7 x - 15)}^{\frac{1}{2}}} (- 10)}{2 {(7 x - 15)}^{\frac{1}{2}}}$

Step 24.4.4

Factor $3 e^{x {(7 x - 15)}^{\frac{1}{2}}}$ out of $3 e^{x {(7 x - 15)}^{\frac{1}{2}}} (7 x) + 3 e^{x {(7 x - 15)}^{\frac{1}{2}}} (- 10)$ .

$\frac{3 e^{x {(7 x - 15)}^{\frac{1}{2}}} (7 x - 10)}{2 {(7 x - 15)}^{\frac{1}{2}}}$