Calculus Examples

$y = (\sqrt{2 x + 5}) \tan (x^{2} + 5 x)$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x + 5}$ as ${(2 x + 5)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(2 x + 5)}^{\frac{1}{2}} \tan (x^{2} + 5 x)]$

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 $f (x) = {(2 x + 5)}^{\frac{1}{2}}$ and $g (x) = \tan (x^{2} + 5 x)$ .

${(2 x + 5)}^{\frac{1}{2}} \frac{d}{d x} [\tan (x^{2} + 5 x)] + \tan (x^{2} + 5 x) \frac{d}{d x} [{(2 x + 5)}^{\frac{1}{2}}]$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \tan (x)$ and $g (x) = x^{2} + 5 x$ .

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Step 3.1

To apply the Chain Rule, set $u_{1}$ as $x^{2} + 5 x$ .

${(2 x + 5)}^{\frac{1}{2}} (\frac{d}{d u_{1}} [\tan (u_{1})] \frac{d}{d x} [x^{2} + 5 x]) + \tan (x^{2} + 5 x) \frac{d}{d x} [{(2 x + 5)}^{\frac{1}{2}}]$

Step 3.2

The derivative of $\tan (u_{1})$ with respect to $u_{1}$ is $\sec^{2} (u_{1})$ .

${(2 x + 5)}^{\frac{1}{2}} (\sec^{2} (u_{1}) \frac{d}{d x} [x^{2} + 5 x]) + \tan (x^{2} + 5 x) \frac{d}{d x} [{(2 x + 5)}^{\frac{1}{2}}]$

Step 3.3

Replace all occurrences of $u_{1}$ with $x^{2} + 5 x$ .

${(2 x + 5)}^{\frac{1}{2}} (\sec^{2} (x^{2} + 5 x) \frac{d}{d x} [x^{2} + 5 x]) + \tan (x^{2} + 5 x) \frac{d}{d x} [{(2 x + 5)}^{\frac{1}{2}}]$

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $x^{2} + 5 x$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [5 x]$ .

${(2 x + 5)}^{\frac{1}{2}} \sec^{2} (x^{2} + 5 x) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [5 x]) + \tan (x^{2} + 5 x) \frac{d}{d x} [{(2 x + 5)}^{\frac{1}{2}}]$

Step 4.2

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

${(2 x + 5)}^{\frac{1}{2}} \sec^{2} (x^{2} + 5 x) (2 x + \frac{d}{d x} [5 x]) + \tan (x^{2} + 5 x) \frac{d}{d x} [{(2 x + 5)}^{\frac{1}{2}}]$

Step 4.3

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

${(2 x + 5)}^{\frac{1}{2}} \sec^{2} (x^{2} + 5 x) (2 x + 5 \frac{d}{d x} [x]) + \tan (x^{2} + 5 x) \frac{d}{d x} [{(2 x + 5)}^{\frac{1}{2}}]$

Step 4.4

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

${(2 x + 5)}^{\frac{1}{2}} \sec^{2} (x^{2} + 5 x) (2 x + 5 \cdot 1) + \tan (x^{2} + 5 x) \frac{d}{d x} [{(2 x + 5)}^{\frac{1}{2}}]$

Step 4.5

Multiply $5$ by $1$ .

${(2 x + 5)}^{\frac{1}{2}} \sec^{2} (x^{2} + 5 x) (2 x + 5) + \tan (x^{2} + 5 x) \frac{d}{d x} [{(2 x + 5)}^{\frac{1}{2}}]$

Step 5

Raise $2 x + 5$ to the power of $1$ .

${(2 x + 5)}^{1} {(2 x + 5)}^{\frac{1}{2}} \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x) \frac{d}{d x} [{(2 x + 5)}^{\frac{1}{2}}]$

Step 6

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

${(2 x + 5)}^{1 + \frac{1}{2}} \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x) \frac{d}{d x} [{(2 x + 5)}^{\frac{1}{2}}]$

Step 7

Simplify the expression.

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Step 7.1

Write $1$ as a fraction with a common denominator.

${(2 x + 5)}^{\frac{2}{2} + \frac{1}{2}} \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x) \frac{d}{d x} [{(2 x + 5)}^{\frac{1}{2}}]$

Step 7.2

Combine the numerators over the common denominator.

${(2 x + 5)}^{\frac{2 + 1}{2}} \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x) \frac{d}{d x} [{(2 x + 5)}^{\frac{1}{2}}]$

Step 7.3

Add $2$ and $1$ .

${(2 x + 5)}^{\frac{3}{2}} \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x) \frac{d}{d x} [{(2 x + 5)}^{\frac{1}{2}}]$

Step 8

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 $g (x) = 2 x + 5$ .

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Step 8.1

To apply the Chain Rule, set $u_{2}$ as $2 x + 5$ .

${(2 x + 5)}^{\frac{3}{2}} \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x) (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{2}}] \frac{d}{d x} [2 x + 5])$

Step 8.2

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

${(2 x + 5)}^{\frac{3}{2}} \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x) (\frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d x} [2 x + 5])$

Step 8.3

Replace all occurrences of $u_{2}$ with $2 x + 5$ .

${(2 x + 5)}^{\frac{3}{2}} \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x) (\frac{1}{2} {(2 x + 5)}^{\frac{1}{2} - 1} \frac{d}{d x} [2 x + 5])$

Step 9

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

${(2 x + 5)}^{\frac{3}{2}} \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x) (\frac{1}{2} {(2 x + 5)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [2 x + 5])$

Step 10

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

${(2 x + 5)}^{\frac{3}{2}} \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x) (\frac{1}{2} {(2 x + 5)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [2 x + 5])$

Step 11

Combine the numerators over the common denominator.

${(2 x + 5)}^{\frac{3}{2}} \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x) (\frac{1}{2} {(2 x + 5)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [2 x + 5])$

Step 12

Simplify the numerator.

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Step 12.1

Multiply $- 1$ by $2$ .

${(2 x + 5)}^{\frac{3}{2}} \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x) (\frac{1}{2} {(2 x + 5)}^{\frac{1 - 2}{2}} \frac{d}{d x} [2 x + 5])$

Step 12.2

Subtract $2$ from $1$ .

${(2 x + 5)}^{\frac{3}{2}} \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x) (\frac{1}{2} {(2 x + 5)}^{\frac{- 1}{2}} \frac{d}{d x} [2 x + 5])$

Step 13

Combine fractions.

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Step 13.1

Move the negative in front of the fraction.

${(2 x + 5)}^{\frac{3}{2}} \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x) (\frac{1}{2} {(2 x + 5)}^{- \frac{1}{2}} \frac{d}{d x} [2 x + 5])$

Step 13.2

Combine $\frac{1}{2}$ and ${(2 x + 5)}^{- \frac{1}{2}}$ .

${(2 x + 5)}^{\frac{3}{2}} \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x) (\frac{{(2 x + 5)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [2 x + 5])$

Step 13.3

Move ${(2 x + 5)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(2 x + 5)}^{\frac{3}{2}} \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x) (\frac{1}{2 {(2 x + 5)}^{\frac{1}{2}}} \frac{d}{d x} [2 x + 5])$

Step 13.4

Combine $\frac{1}{2 {(2 x + 5)}^{\frac{1}{2}}}$ and $\tan (x^{2} + 5 x)$ .

${(2 x + 5)}^{\frac{3}{2}} \sec^{2} (x^{2} + 5 x) + \frac{\tan (x^{2} + 5 x)}{2 {(2 x + 5)}^{\frac{1}{2}}} \frac{d}{d x} [2 x + 5]$

Step 14

By the Sum Rule, the derivative of $2 x + 5$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [5]$ .

${(2 x + 5)}^{\frac{3}{2}} \sec^{2} (x^{2} + 5 x) + \frac{\tan (x^{2} + 5 x)}{2 {(2 x + 5)}^{\frac{1}{2}}} (\frac{d}{d x} [2 x] + \frac{d}{d x} [5])$

Step 15

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

${(2 x + 5)}^{\frac{3}{2}} \sec^{2} (x^{2} + 5 x) + \frac{\tan (x^{2} + 5 x)}{2 {(2 x + 5)}^{\frac{1}{2}}} (2 \frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 16

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

${(2 x + 5)}^{\frac{3}{2}} \sec^{2} (x^{2} + 5 x) + \frac{\tan (x^{2} + 5 x)}{2 {(2 x + 5)}^{\frac{1}{2}}} (2 \cdot 1 + \frac{d}{d x} [5])$

Step 17

Multiply $2$ by $1$ .

${(2 x + 5)}^{\frac{3}{2}} \sec^{2} (x^{2} + 5 x) + \frac{\tan (x^{2} + 5 x)}{2 {(2 x + 5)}^{\frac{1}{2}}} (2 + \frac{d}{d x} [5])$

Step 18

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

${(2 x + 5)}^{\frac{3}{2}} \sec^{2} (x^{2} + 5 x) + \frac{\tan (x^{2} + 5 x)}{2 {(2 x + 5)}^{\frac{1}{2}}} (2 + 0)$

Step 19

Simplify terms.

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Step 19.1

Add $2$ and $0$ .

${(2 x + 5)}^{\frac{3}{2}} \sec^{2} (x^{2} + 5 x) + \frac{\tan (x^{2} + 5 x)}{2 {(2 x + 5)}^{\frac{1}{2}}} \cdot 2$

Step 19.2

Combine $\frac{\tan (x^{2} + 5 x)}{2 {(2 x + 5)}^{\frac{1}{2}}}$ and $2$ .

${(2 x + 5)}^{\frac{3}{2}} \sec^{2} (x^{2} + 5 x) + \frac{\tan (x^{2} + 5 x) \cdot 2}{2 {(2 x + 5)}^{\frac{1}{2}}}$

Step 19.3

Move $2$ to the left of $\tan (x^{2} + 5 x)$ .

${(2 x + 5)}^{\frac{3}{2}} \sec^{2} (x^{2} + 5 x) + \frac{2 \cdot \tan (x^{2} + 5 x)}{2 {(2 x + 5)}^{\frac{1}{2}}}$

Step 19.4

Cancel the common factor.

${(2 x + 5)}^{\frac{3}{2}} \sec^{2} (x^{2} + 5 x) + \frac{2 \tan (x^{2} + 5 x)}{2 {(2 x + 5)}^{\frac{1}{2}}}$

Step 19.5

Rewrite the expression.

${(2 x + 5)}^{\frac{3}{2}} \sec^{2} (x^{2} + 5 x) + \frac{\tan (x^{2} + 5 x)}{{(2 x + 5)}^{\frac{1}{2}}}$

Step 20

To write ${(2 x + 5)}^{\frac{3}{2}} \sec^{2} (x^{2} + 5 x)$ as a fraction with a common denominator, multiply by $\frac{{(2 x + 5)}^{\frac{1}{2}}}{{(2 x + 5)}^{\frac{1}{2}}}$ .

$\frac{{(2 x + 5)}^{\frac{3}{2}} \sec^{2} (x^{2} + 5 x) {(2 x + 5)}^{\frac{1}{2}}}{{(2 x + 5)}^{\frac{1}{2}}} + \frac{\tan (x^{2} + 5 x)}{{(2 x + 5)}^{\frac{1}{2}}}$

Step 21

Combine the numerators over the common denominator.

$\frac{{(2 x + 5)}^{\frac{3}{2}} \sec^{2} (x^{2} + 5 x) {(2 x + 5)}^{\frac{1}{2}} + \tan (x^{2} + 5 x)}{{(2 x + 5)}^{\frac{1}{2}}}$

Step 22

Multiply ${(2 x + 5)}^{\frac{3}{2}}$ by ${(2 x + 5)}^{\frac{1}{2}}$ by adding the exponents.

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Step 22.1

Move ${(2 x + 5)}^{\frac{1}{2}}$ .

$\frac{{(2 x + 5)}^{\frac{1}{2}} {(2 x + 5)}^{\frac{3}{2}} \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x)}{{(2 x + 5)}^{\frac{1}{2}}}$

Step 22.2

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

$\frac{{(2 x + 5)}^{\frac{1}{2} + \frac{3}{2}} \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x)}{{(2 x + 5)}^{\frac{1}{2}}}$

Step 22.3

Combine the numerators over the common denominator.

$\frac{{(2 x + 5)}^{\frac{1 + 3}{2}} \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x)}{{(2 x + 5)}^{\frac{1}{2}}}$

Step 22.4

Add $1$ and $3$ .

$\frac{{(2 x + 5)}^{\frac{4}{2}} \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x)}{{(2 x + 5)}^{\frac{1}{2}}}$

Step 22.5

Divide $4$ by $2$ .

$\frac{{(2 x + 5)}^{2} \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x)}{{(2 x + 5)}^{\frac{1}{2}}}$

Step 23

Simplify each term.

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Step 23.1

Rewrite ${(2 x + 5)}^{2}$ as $(2 x + 5) (2 x + 5)$ .

$\frac{(2 x + 5) (2 x + 5) \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x)}{{(2 x + 5)}^{\frac{1}{2}}}$

Step 23.2

Expand $(2 x + 5) (2 x + 5)$ using the FOIL Method.

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Step 23.2.1

Apply the distributive property.

$\frac{(2 x (2 x + 5) + 5 (2 x + 5)) \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x)}{{(2 x + 5)}^{\frac{1}{2}}}$

Step 23.2.2

Apply the distributive property.

$\frac{(2 x (2 x) + 2 x \cdot 5 + 5 (2 x + 5)) \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x)}{{(2 x + 5)}^{\frac{1}{2}}}$

Step 23.2.3

Apply the distributive property.

$\frac{(2 x (2 x) + 2 x \cdot 5 + 5 (2 x) + 5 \cdot 5) \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x)}{{(2 x + 5)}^{\frac{1}{2}}}$

Step 23.3

Simplify and combine like terms.

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Step 23.3.1

Simplify each term.

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Step 23.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{(2 \cdot 2 x \cdot x + 2 x \cdot 5 + 5 (2 x) + 5 \cdot 5) \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x)}{{(2 x + 5)}^{\frac{1}{2}}}$

Step 23.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 23.3.1.2.1

Move $x$ .

$\frac{(2 \cdot 2 (x \cdot x) + 2 x \cdot 5 + 5 (2 x) + 5 \cdot 5) \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x)}{{(2 x + 5)}^{\frac{1}{2}}}$

Step 23.3.1.2.2

Multiply $x$ by $x$ .

$\frac{(2 \cdot 2 x^{2} + 2 x \cdot 5 + 5 (2 x) + 5 \cdot 5) \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x)}{{(2 x + 5)}^{\frac{1}{2}}}$

Step 23.3.1.3

Multiply $2$ by $2$ .

$\frac{(4 x^{2} + 2 x \cdot 5 + 5 (2 x) + 5 \cdot 5) \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x)}{{(2 x + 5)}^{\frac{1}{2}}}$

Step 23.3.1.4

Multiply $5$ by $2$ .

$\frac{(4 x^{2} + 10 x + 5 (2 x) + 5 \cdot 5) \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x)}{{(2 x + 5)}^{\frac{1}{2}}}$

Step 23.3.1.5

Multiply $2$ by $5$ .

$\frac{(4 x^{2} + 10 x + 10 x + 5 \cdot 5) \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x)}{{(2 x + 5)}^{\frac{1}{2}}}$

Step 23.3.1.6

Multiply $5$ by $5$ .

$\frac{(4 x^{2} + 10 x + 10 x + 25) \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x)}{{(2 x + 5)}^{\frac{1}{2}}}$

Step 23.3.2

Add $10 x$ and $10 x$ .

$\frac{(4 x^{2} + 20 x + 25) \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x)}{{(2 x + 5)}^{\frac{1}{2}}}$

Step 23.4

Apply the distributive property.

$\frac{4 x^{2} \sec^{2} (x^{2} + 5 x) + 20 x \sec^{2} (x^{2} + 5 x) + 25 \sec^{2} (x^{2} + 5 x) + \tan (x^{2} + 5 x)}{{(2 x + 5)}^{\frac{1}{2}}}$