Algebra Examples

$y = 16^{4} \sqrt{4 x^{4} + 4}$

Step 1

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

$y = 16^{4} {(4 x^{4} + 4)}^{\frac{1}{2}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d y} (y) = \frac{d}{d y} (16^{4} {(4 x^{4} + 4)}^{\frac{1}{2}})$

Step 3

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

$1$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Differentiate using the Constant Multiple Rule.

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Step 4.1.1

Raise $16$ to the power of $4$ .

$\frac{d}{d y} [65536 {(4 x^{4} + 4)}^{\frac{1}{2}}]$

Step 4.1.2

Since $65536$ is constant with respect to $y$ , the derivative of $65536 {(4 x^{4} + 4)}^{\frac{1}{2}}$ with respect to $y$ is $65536 \frac{d}{d y} [{(4 x^{4} + 4)}^{\frac{1}{2}}]$ .

$65536 \frac{d}{d y} [{(4 x^{4} + 4)}^{\frac{1}{2}}]$

Step 4.2

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

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Step 4.2.1

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

$65536 (\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{2}}] \frac{d}{d y} [4 x^{4} + 4])$

Step 4.2.2

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

$65536 (\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d y} [4 x^{4} + 4])$

Step 4.2.3

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

$65536 (\frac{1}{2} {(4 x^{4} + 4)}^{\frac{1}{2} - 1} \frac{d}{d y} [4 x^{4} + 4])$

Step 4.3

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

$65536 (\frac{1}{2} {(4 x^{4} + 4)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d y} [4 x^{4} + 4])$

Step 4.4

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

$65536 (\frac{1}{2} {(4 x^{4} + 4)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d y} [4 x^{4} + 4])$

Step 4.5

Combine the numerators over the common denominator.

$65536 (\frac{1}{2} {(4 x^{4} + 4)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d y} [4 x^{4} + 4])$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $2$ .

$65536 (\frac{1}{2} {(4 x^{4} + 4)}^{\frac{1 - 2}{2}} \frac{d}{d y} [4 x^{4} + 4])$

Step 4.6.2

Subtract $2$ from $1$ .

$65536 (\frac{1}{2} {(4 x^{4} + 4)}^{\frac{- 1}{2}} \frac{d}{d y} [4 x^{4} + 4])$

Step 4.7

Move the negative in front of the fraction.

$65536 (\frac{1}{2} {(4 x^{4} + 4)}^{- \frac{1}{2}} \frac{d}{d y} [4 x^{4} + 4])$

Step 4.8

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

$65536 (\frac{{(4 x^{4} + 4)}^{- \frac{1}{2}}}{2} \frac{d}{d y} [4 x^{4} + 4])$

Step 4.9

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

$65536 (\frac{1}{2 {(4 x^{4} + 4)}^{\frac{1}{2}}} \frac{d}{d y} [4 x^{4} + 4])$

Step 4.10

Combine $\frac{1}{2 {(4 x^{4} + 4)}^{\frac{1}{2}}}$ and $65536$ .

$\frac{65536}{2 {(4 x^{4} + 4)}^{\frac{1}{2}}} \frac{d}{d y} [4 x^{4} + 4]$

Step 4.11

Factor $2$ out of $65536$ .

$\frac{2 \cdot 32768}{2 {(4 x^{4} + 4)}^{\frac{1}{2}}} \frac{d}{d y} [4 x^{4} + 4]$

Step 4.12

Cancel the common factors.

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Step 4.12.1

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

$\frac{2 \cdot 32768}{2 ({(4 x^{4} + 4)}^{\frac{1}{2}})} \frac{d}{d y} [4 x^{4} + 4]$

Step 4.12.2

Cancel the common factor.

$\frac{2 \cdot 32768}{2 {(4 x^{4} + 4)}^{\frac{1}{2}}} \frac{d}{d y} [4 x^{4} + 4]$

Step 4.12.3

Rewrite the expression.

$\frac{32768}{{(4 x^{4} + 4)}^{\frac{1}{2}}} \frac{d}{d y} [4 x^{4} + 4]$

Step 4.13

Differentiate.

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Step 4.13.1

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

$\frac{32768}{{(4 x^{4} + 4)}^{\frac{1}{2}}} (\frac{d}{d y} [4 x^{4}] + \frac{d}{d y} [4])$

Step 4.13.2

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

$\frac{32768}{{(4 x^{4} + 4)}^{\frac{1}{2}}} (4 \frac{d}{d y} [x^{4}] + \frac{d}{d y} [4])$

Step 4.14

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

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Step 4.14.1

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

$\frac{32768}{{(4 x^{4} + 4)}^{\frac{1}{2}}} (4 (\frac{d}{d u_{2}} [{u_{2}}^{4}] \frac{d}{d y} [x]) + \frac{d}{d y} [4])$

Step 4.14.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 4$ .

$\frac{32768}{{(4 x^{4} + 4)}^{\frac{1}{2}}} (4 (4 {u_{2}}^{3} \frac{d}{d y} [x]) + \frac{d}{d y} [4])$

Step 4.14.3

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

$\frac{32768}{{(4 x^{4} + 4)}^{\frac{1}{2}}} (4 (4 x^{3} \frac{d}{d y} [x]) + \frac{d}{d y} [4])$

Step 4.15

Multiply $4$ by $4$ .

$\frac{32768}{{(4 x^{4} + 4)}^{\frac{1}{2}}} (16 (x^{3} \frac{d}{d y} [x]) + \frac{d}{d y} [4])$

Step 4.16

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

$\frac{32768}{{(4 x^{4} + 4)}^{\frac{1}{2}}} (16 x^{3} x' + \frac{d}{d y} [4])$

Step 4.17

Since $4$ is constant with respect to $y$ , the derivative of $4$ with respect to $y$ is $0$ .

$\frac{32768}{{(4 x^{4} + 4)}^{\frac{1}{2}}} (16 x^{3} x' + 0)$

Step 4.18

Combine fractions.

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Step 4.18.1

Add $16 x^{3} x'$ and $0$ .

$\frac{32768}{{(4 x^{4} + 4)}^{\frac{1}{2}}} (16 x^{3} x')$

Step 4.18.2

Combine $16$ and $\frac{32768}{{(4 x^{4} + 4)}^{\frac{1}{2}}}$ .

$\frac{16 \cdot 32768}{{(4 x^{4} + 4)}^{\frac{1}{2}}} (x^{3} x')$

Step 4.18.3

Multiply $16$ by $32768$ .

$\frac{524288}{{(4 x^{4} + 4)}^{\frac{1}{2}}} (x^{3} x')$

Step 4.18.4

Combine $x^{3}$ and $\frac{524288}{{(4 x^{4} + 4)}^{\frac{1}{2}}}$ .

$\frac{x^{3} \cdot 524288}{{(4 x^{4} + 4)}^{\frac{1}{2}}} x'$

Step 4.18.5

Combine $\frac{x^{3} \cdot 524288}{{(4 x^{4} + 4)}^{\frac{1}{2}}}$ and $x'$ .

$\frac{x^{3} \cdot 524288 x'}{{(4 x^{4} + 4)}^{\frac{1}{2}}}$

Step 4.18.6

Move $524288$ to the left of $x^{3}$ .

$\frac{524288 x^{3} x'}{{(4 x^{4} + 4)}^{\frac{1}{2}}}$

Step 5

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

$1 = \frac{524288 x^{3} x'}{{(4 x^{4} + 4)}^{\frac{1}{2}}}$

Step 6

Solve for $x'$ .

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Step 6.1

Rewrite the equation as $\frac{524288 x^{3} x'}{{(4 x^{4} + 4)}^{\frac{1}{2}}} = 1$ .

$\frac{524288 x^{3} x'}{{(4 x^{4} + 4)}^{\frac{1}{2}}} = 1$

Step 6.2

Multiply both sides by ${(4 x^{4} + 4)}^{\frac{1}{2}}$ .

$\frac{524288 x^{3} x'}{{(4 x^{4} + 4)}^{\frac{1}{2}}} {(4 x^{4} + 4)}^{\frac{1}{2}} = 1 {(4 x^{4} + 4)}^{\frac{1}{2}}$

Step 6.3

Simplify.

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Step 6.3.1

Simplify the left side.

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Step 6.3.1.1

Cancel the common factor of ${(4 x^{4} + 4)}^{\frac{1}{2}}$ .

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Step 6.3.1.1.1

Cancel the common factor.

$\frac{524288 x^{3} x'}{{(4 x^{4} + 4)}^{\frac{1}{2}}} {(4 x^{4} + 4)}^{\frac{1}{2}} = 1 {(4 x^{4} + 4)}^{\frac{1}{2}}$

Step 6.3.1.1.2

Rewrite the expression.

$524288 x^{3} x' = 1 {(4 x^{4} + 4)}^{\frac{1}{2}}$

Step 6.3.2

Simplify the right side.

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Step 6.3.2.1

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

$524288 x^{3} x' = {(4 x^{4} + 4)}^{\frac{1}{2}}$

Step 6.4

Divide each term in $524288 x^{3} x' = {(4 x^{4} + 4)}^{\frac{1}{2}}$ by $524288 x^{3}$ and simplify.

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Step 6.4.1

Divide each term in $524288 x^{3} x' = {(4 x^{4} + 4)}^{\frac{1}{2}}$ by $524288 x^{3}$ .

$\frac{524288 x^{3} x'}{524288 x^{3}} = \frac{{(4 x^{4} + 4)}^{\frac{1}{2}}}{524288 x^{3}}$

Step 6.4.2

Simplify the left side.

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Step 6.4.2.1

Cancel the common factor of $524288$ .

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Step 6.4.2.1.1

Cancel the common factor.

$\frac{524288 x^{3} x'}{524288 x^{3}} = \frac{{(4 x^{4} + 4)}^{\frac{1}{2}}}{524288 x^{3}}$

Step 6.4.2.1.2

Rewrite the expression.

$\frac{x^{3} x'}{x^{3}} = \frac{{(4 x^{4} + 4)}^{\frac{1}{2}}}{524288 x^{3}}$

Step 6.4.2.2

Cancel the common factor of $x^{3}$ .

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Step 6.4.2.2.1

Cancel the common factor.

$\frac{x^{3} x'}{x^{3}} = \frac{{(4 x^{4} + 4)}^{\frac{1}{2}}}{524288 x^{3}}$

Step 6.4.2.2.2

Divide $x'$ by $1$ .

$x' = \frac{{(4 x^{4} + 4)}^{\frac{1}{2}}}{524288 x^{3}}$

Step 7

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

$\frac{d x}{d y} = \frac{{(4 x^{4} + 4)}^{\frac{1}{2}}}{524288 x^{3}}$