Calculus Examples

$y = - 2 x^{4} \sqrt{4 x + 5}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x + 5}$ as ${(4 x + 5)}^{\frac{1}{2}}$ .

$y = - 2 x^{4} {(4 x + 5)}^{\frac{1}{2}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (- 2 x^{4} {(4 x + 5)}^{\frac{1}{2}})$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{4} {(4 x + 5)}^{\frac{1}{2}}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{4} {(4 x + 5)}^{\frac{1}{2}}]$ .

$- 2 \frac{d}{d x} [x^{4} {(4 x + 5)}^{\frac{1}{2}}]$

Step 4.2

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 $f (x) = x^{4}$ and $g (x) = {(4 x + 5)}^{\frac{1}{2}}$ .

$- 2 (x^{4} \frac{d}{d x} [{(4 x + 5)}^{\frac{1}{2}}] + {(4 x + 5)}^{\frac{1}{2}} \frac{d}{d x} [x^{4}])$

Step 4.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 4 x + 5$ .

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Step 4.3.1

To apply the Chain Rule, set $u$ as $4 x + 5$ .

$- 2 (x^{4} (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [4 x + 5]) + {(4 x + 5)}^{\frac{1}{2}} \frac{d}{d x} [x^{4}])$

Step 4.3.2

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

$- 2 (x^{4} (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [4 x + 5]) + {(4 x + 5)}^{\frac{1}{2}} \frac{d}{d x} [x^{4}])$

Step 4.3.3

Replace all occurrences of $u$ with $4 x + 5$ .

$- 2 (x^{4} (\frac{1}{2} {(4 x + 5)}^{\frac{1}{2} - 1} \frac{d}{d x} [4 x + 5]) + {(4 x + 5)}^{\frac{1}{2}} \frac{d}{d x} [x^{4}])$

Step 4.4

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

$- 2 (x^{4} (\frac{1}{2} {(4 x + 5)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [4 x + 5]) + {(4 x + 5)}^{\frac{1}{2}} \frac{d}{d x} [x^{4}])$

Step 4.5

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

$- 2 (x^{4} (\frac{1}{2} {(4 x + 5)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [4 x + 5]) + {(4 x + 5)}^{\frac{1}{2}} \frac{d}{d x} [x^{4}])$

Step 4.6

Combine the numerators over the common denominator.

$- 2 (x^{4} (\frac{1}{2} {(4 x + 5)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [4 x + 5]) + {(4 x + 5)}^{\frac{1}{2}} \frac{d}{d x} [x^{4}])$

Step 4.7

Simplify the numerator.

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Step 4.7.1

Multiply $- 1$ by $2$ .

$- 2 (x^{4} (\frac{1}{2} {(4 x + 5)}^{\frac{1 - 2}{2}} \frac{d}{d x} [4 x + 5]) + {(4 x + 5)}^{\frac{1}{2}} \frac{d}{d x} [x^{4}])$

Step 4.7.2

Subtract $2$ from $1$ .

$- 2 (x^{4} (\frac{1}{2} {(4 x + 5)}^{\frac{- 1}{2}} \frac{d}{d x} [4 x + 5]) + {(4 x + 5)}^{\frac{1}{2}} \frac{d}{d x} [x^{4}])$

Step 4.8

Combine fractions.

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Step 4.8.1

Move the negative in front of the fraction.

$- 2 (x^{4} (\frac{1}{2} {(4 x + 5)}^{- \frac{1}{2}} \frac{d}{d x} [4 x + 5]) + {(4 x + 5)}^{\frac{1}{2}} \frac{d}{d x} [x^{4}])$

Step 4.8.2

Combine $\frac{1}{2}$ and ${(4 x + 5)}^{- \frac{1}{2}}$ .

$- 2 (x^{4} (\frac{{(4 x + 5)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [4 x + 5]) + {(4 x + 5)}^{\frac{1}{2}} \frac{d}{d x} [x^{4}])$

Step 4.8.3

Move ${(4 x + 5)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 2 (x^{4} (\frac{1}{2 {(4 x + 5)}^{\frac{1}{2}}} \frac{d}{d x} [4 x + 5]) + {(4 x + 5)}^{\frac{1}{2}} \frac{d}{d x} [x^{4}])$

Step 4.8.4

Combine $\frac{1}{2 {(4 x + 5)}^{\frac{1}{2}}}$ and $x^{4}$ .

$- 2 (\frac{x^{4}}{2 {(4 x + 5)}^{\frac{1}{2}}} \frac{d}{d x} [4 x + 5] + {(4 x + 5)}^{\frac{1}{2}} \frac{d}{d x} [x^{4}])$

Step 4.9

By the Sum Rule, the derivative of $4 x + 5$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [5]$ .

$- 2 (\frac{x^{4}}{2 {(4 x + 5)}^{\frac{1}{2}}} (\frac{d}{d x} [4 x] + \frac{d}{d x} [5]) + {(4 x + 5)}^{\frac{1}{2}} \frac{d}{d x} [x^{4}])$

Step 4.10

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

$- 2 (\frac{x^{4}}{2 {(4 x + 5)}^{\frac{1}{2}}} (4 \frac{d}{d x} [x] + \frac{d}{d x} [5]) + {(4 x + 5)}^{\frac{1}{2}} \frac{d}{d x} [x^{4}])$

Step 4.11

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

$- 2 (\frac{x^{4}}{2 {(4 x + 5)}^{\frac{1}{2}}} (4 \cdot 1 + \frac{d}{d x} [5]) + {(4 x + 5)}^{\frac{1}{2}} \frac{d}{d x} [x^{4}])$

Step 4.12

Multiply $4$ by $1$ .

$- 2 (\frac{x^{4}}{2 {(4 x + 5)}^{\frac{1}{2}}} (4 + \frac{d}{d x} [5]) + {(4 x + 5)}^{\frac{1}{2}} \frac{d}{d x} [x^{4}])$

Step 4.13

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

$- 2 (\frac{x^{4}}{2 {(4 x + 5)}^{\frac{1}{2}}} (4 + 0) + {(4 x + 5)}^{\frac{1}{2}} \frac{d}{d x} [x^{4}])$

Step 4.14

Simplify terms.

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Step 4.14.1

Add $4$ and $0$ .

$- 2 (\frac{x^{4}}{2 {(4 x + 5)}^{\frac{1}{2}}} \cdot 4 + {(4 x + 5)}^{\frac{1}{2}} \frac{d}{d x} [x^{4}])$

Step 4.14.2

Combine $\frac{x^{4}}{2 {(4 x + 5)}^{\frac{1}{2}}}$ and $4$ .

$- 2 (\frac{x^{4} \cdot 4}{2 {(4 x + 5)}^{\frac{1}{2}}} + {(4 x + 5)}^{\frac{1}{2}} \frac{d}{d x} [x^{4}])$

Step 4.14.3

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

$- 2 (\frac{4 \cdot x^{4}}{2 {(4 x + 5)}^{\frac{1}{2}}} + {(4 x + 5)}^{\frac{1}{2}} \frac{d}{d x} [x^{4}])$

Step 4.14.4

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

$- 2 (\frac{2 (2 x^{4})}{2 {(4 x + 5)}^{\frac{1}{2}}} + {(4 x + 5)}^{\frac{1}{2}} \frac{d}{d x} [x^{4}])$

Step 4.15

Cancel the common factors.

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Step 4.15.1

Factor $2$ out of $2 {(4 x + 5)}^{\frac{1}{2}}$ .

$- 2 (\frac{2 (2 x^{4})}{2 ({(4 x + 5)}^{\frac{1}{2}})} + {(4 x + 5)}^{\frac{1}{2}} \frac{d}{d x} [x^{4}])$

Step 4.15.2

Cancel the common factor.

$- 2 (\frac{2 (2 x^{4})}{2 {(4 x + 5)}^{\frac{1}{2}}} + {(4 x + 5)}^{\frac{1}{2}} \frac{d}{d x} [x^{4}])$

Step 4.15.3

Rewrite the expression.

$- 2 (\frac{2 x^{4}}{{(4 x + 5)}^{\frac{1}{2}}} + {(4 x + 5)}^{\frac{1}{2}} \frac{d}{d x} [x^{4}])$

Step 4.16

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$- 2 (\frac{2 x^{4}}{{(4 x + 5)}^{\frac{1}{2}}} + {(4 x + 5)}^{\frac{1}{2}} (4 x^{3}))$

Step 4.17

Move $4$ to the left of ${(4 x + 5)}^{\frac{1}{2}}$ .

$- 2 (\frac{2 x^{4}}{{(4 x + 5)}^{\frac{1}{2}}} + 4 \cdot {(4 x + 5)}^{\frac{1}{2}} x^{3})$

Step 4.18

Combine $\frac{2 x^{4}}{{(4 x + 5)}^{\frac{1}{2}}}$ and $4 {(4 x + 5)}^{\frac{1}{2}} x^{3}$ using a common denominator.

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Step 4.18.1

Move ${(4 x + 5)}^{\frac{1}{2}}$ .

$- 2 (\frac{2 x^{4}}{{(4 x + 5)}^{\frac{1}{2}}} + 4 x^{3} {(4 x + 5)}^{\frac{1}{2}})$

Step 4.18.2

To write $4 x^{3} {(4 x + 5)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(4 x + 5)}^{\frac{1}{2}}}{{(4 x + 5)}^{\frac{1}{2}}}$ .

$- 2 (\frac{2 x^{4}}{{(4 x + 5)}^{\frac{1}{2}}} + \frac{4 x^{3} {(4 x + 5)}^{\frac{1}{2}} {(4 x + 5)}^{\frac{1}{2}}}{{(4 x + 5)}^{\frac{1}{2}}})$

Step 4.18.3

Combine the numerators over the common denominator.

$- 2 \frac{2 x^{4} + 4 x^{3} {(4 x + 5)}^{\frac{1}{2}} {(4 x + 5)}^{\frac{1}{2}}}{{(4 x + 5)}^{\frac{1}{2}}}$

Step 4.19

Multiply ${(4 x + 5)}^{\frac{1}{2}}$ by ${(4 x + 5)}^{\frac{1}{2}}$ by adding the exponents.

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Step 4.19.1

Move ${(4 x + 5)}^{\frac{1}{2}}$ .

$- 2 \frac{2 x^{4} + 4 x^{3} ({(4 x + 5)}^{\frac{1}{2}} {(4 x + 5)}^{\frac{1}{2}})}{{(4 x + 5)}^{\frac{1}{2}}}$

Step 4.19.2

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

$- 2 \frac{2 x^{4} + 4 x^{3} {(4 x + 5)}^{\frac{1}{2} + \frac{1}{2}}}{{(4 x + 5)}^{\frac{1}{2}}}$

Step 4.19.3

Combine the numerators over the common denominator.

$- 2 \frac{2 x^{4} + 4 x^{3} {(4 x + 5)}^{\frac{1 + 1}{2}}}{{(4 x + 5)}^{\frac{1}{2}}}$

Step 4.19.4

Add $1$ and $1$ .

$- 2 \frac{2 x^{4} + 4 x^{3} {(4 x + 5)}^{\frac{2}{2}}}{{(4 x + 5)}^{\frac{1}{2}}}$

Step 4.19.5

Divide $2$ by $2$ .

$- 2 \frac{2 x^{4} + 4 x^{3} {(4 x + 5)}^{1}}{{(4 x + 5)}^{\frac{1}{2}}}$

Step 4.20

Simplify $4 x^{3} {(4 x + 5)}^{1}$ .

$- 2 \frac{2 x^{4} + 4 x^{3} (4 x + 5)}{{(4 x + 5)}^{\frac{1}{2}}}$

Step 4.21

Combine $- 2$ and $\frac{2 x^{4} + 4 x^{3} (4 x + 5)}{{(4 x + 5)}^{\frac{1}{2}}}$ .

$\frac{- 2 (2 x^{4} + 4 x^{3} (4 x + 5))}{{(4 x + 5)}^{\frac{1}{2}}}$

Step 4.22

Move the negative in front of the fraction.

$- \frac{2 (2 x^{4} + 4 x^{3} (4 x + 5))}{{(4 x + 5)}^{\frac{1}{2}}}$

Step 4.23

Simplify.

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Step 4.23.1

Apply the distributive property.

$- \frac{2 (2 x^{4} + 4 x^{3} (4 x) + 4 x^{3} \cdot 5)}{{(4 x + 5)}^{\frac{1}{2}}}$

Step 4.23.2

Apply the distributive property.

$- \frac{2 (2 x^{4}) + 2 (4 x^{3} (4 x)) + 2 (4 x^{3} \cdot 5)}{{(4 x + 5)}^{\frac{1}{2}}}$

Step 4.23.3

Simplify the numerator.

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Step 4.23.3.1

Simplify each term.

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Step 4.23.3.1.1

Multiply $2$ by $2$ .

$- \frac{4 x^{4} + 2 (4 x^{3} (4 x)) + 2 (4 x^{3} \cdot 5)}{{(4 x + 5)}^{\frac{1}{2}}}$

Step 4.23.3.1.2

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 4.23.3.1.2.1

Move $x$ .

$- \frac{4 x^{4} + 2 (4 (x \cdot x^{3}) \cdot 4) + 2 (4 x^{3} \cdot 5)}{{(4 x + 5)}^{\frac{1}{2}}}$

Step 4.23.3.1.2.2

Multiply $x$ by $x^{3}$ .

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Step 4.23.3.1.2.2.1

Raise $x$ to the power of $1$ .

$- \frac{4 x^{4} + 2 (4 (x^{1} x^{3}) \cdot 4) + 2 (4 x^{3} \cdot 5)}{{(4 x + 5)}^{\frac{1}{2}}}$

Step 4.23.3.1.2.2.2

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

$- \frac{4 x^{4} + 2 (4 x^{1 + 3} \cdot 4) + 2 (4 x^{3} \cdot 5)}{{(4 x + 5)}^{\frac{1}{2}}}$

Step 4.23.3.1.2.3

Add $1$ and $3$ .

$- \frac{4 x^{4} + 2 (4 x^{4} \cdot 4) + 2 (4 x^{3} \cdot 5)}{{(4 x + 5)}^{\frac{1}{2}}}$

Step 4.23.3.1.3

Multiply $4$ by $4$ .

$- \frac{4 x^{4} + 2 (16 x^{4}) + 2 (4 x^{3} \cdot 5)}{{(4 x + 5)}^{\frac{1}{2}}}$

Step 4.23.3.1.4

Multiply $16$ by $2$ .

$- \frac{4 x^{4} + 32 x^{4} + 2 (4 x^{3} \cdot 5)}{{(4 x + 5)}^{\frac{1}{2}}}$

Step 4.23.3.1.5

Multiply $5$ by $4$ .

$- \frac{4 x^{4} + 32 x^{4} + 2 (20 x^{3})}{{(4 x + 5)}^{\frac{1}{2}}}$

Step 4.23.3.1.6

Multiply $20$ by $2$ .

$- \frac{4 x^{4} + 32 x^{4} + 40 x^{3}}{{(4 x + 5)}^{\frac{1}{2}}}$

Step 4.23.3.2

Add $4 x^{4}$ and $32 x^{4}$ .

$- \frac{36 x^{4} + 40 x^{3}}{{(4 x + 5)}^{\frac{1}{2}}}$

Step 4.23.4

Factor $4 x^{3}$ out of $36 x^{4} + 40 x^{3}$ .

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Step 4.23.4.1

Factor $4 x^{3}$ out of $36 x^{4}$ .

$- \frac{4 x^{3} (9 x) + 40 x^{3}}{{(4 x + 5)}^{\frac{1}{2}}}$

Step 4.23.4.2

Factor $4 x^{3}$ out of $40 x^{3}$ .

$- \frac{4 x^{3} (9 x) + 4 x^{3} (10)}{{(4 x + 5)}^{\frac{1}{2}}}$

Step 4.23.4.3

Factor $4 x^{3}$ out of $4 x^{3} (9 x) + 4 x^{3} (10)$ .

$- \frac{4 x^{3} (9 x + 10)}{{(4 x + 5)}^{\frac{1}{2}}}$

Step 5

Reform the equation by setting the left side equal to the right side.

$y' = - \frac{4 x^{3} (9 x + 10)}{{(4 x + 5)}^{\frac{1}{2}}}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{4 x^{3} (9 x + 10)}{{(4 x + 5)}^{\frac{1}{2}}}$