Algebra Examples

$y = \tan^{-1} (\sqrt{x})$

Step 1

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

$y = \tan^{-1} (x^{\frac{1}{2}})$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d y} (y) = \frac{d}{d y} (\tan^{-1} (x^{\frac{1}{2}}))$

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

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Step 4.1.1

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

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

Step 4.1.2

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

$\frac{1}{1 + {u_{1}}^{2}} \frac{d}{d y} [x^{\frac{1}{2}}]$

Step 4.1.3

Replace all occurrences of $u_{1}$ with $x^{\frac{1}{2}}$ .

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

Step 4.2

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{1 + x^{\frac{1}{2} \cdot 2}} \frac{d}{d y} [x^{\frac{1}{2}}]$

Step 4.2.2

Cancel the common factor of $2$ .

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Step 4.2.2.1

Cancel the common factor.

$\frac{1}{1 + x^{\frac{1}{2} \cdot 2}} \frac{d}{d y} [x^{\frac{1}{2}}]$

Step 4.2.2.2

Rewrite the expression.

$\frac{1}{1 + x^{1}} \frac{d}{d y} [x^{\frac{1}{2}}]$

Step 4.3

Simplify.

$\frac{1}{1 + x} \frac{d}{d y} [x^{\frac{1}{2}}]$

Step 4.4

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 $g (y) = x$ .

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Step 4.4.1

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

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

Step 4.4.2

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

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

Step 4.4.3

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

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

Step 4.5

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

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

Step 4.6

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

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

Step 4.7

Combine the numerators over the common denominator.

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

Step 4.8

Simplify the numerator.

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Step 4.8.1

Multiply $- 1$ by $2$ .

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

Step 4.8.2

Subtract $2$ from $1$ .

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

Step 4.9

Move the negative in front of the fraction.

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

Step 4.10

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

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

Step 4.11

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

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

Step 4.12

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

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

Step 4.13

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

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

Step 4.14

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

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

Step 5

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

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

Step 6

Solve for $x'$ .

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Step 6.1

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

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

Step 6.2

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

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

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

Simplify $\frac{x'}{2 x^{\frac{1}{2}} (1 + x)} (2 x^{\frac{1}{2}} (1 + x))$ .

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Step 6.3.1.1.1

Rewrite using the commutative property of multiplication.

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

Step 6.3.1.1.2

Cancel the common factor of $2$ .

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Step 6.3.1.1.2.1

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

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

Step 6.3.1.1.2.2

Cancel the common factor.

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

Step 6.3.1.1.2.3

Rewrite the expression.

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

Step 6.3.1.1.3

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

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Step 6.3.1.1.3.1

Cancel the common factor.

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

Step 6.3.1.1.3.2

Rewrite the expression.

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

Step 6.3.2

Simplify the right side.

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Step 6.3.2.1

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

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Step 6.3.2.1.1

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

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

Step 6.3.2.1.2

Apply the distributive property.

$x' = 2 x^{\frac{1}{2}} \cdot 1 + 2 x^{\frac{1}{2}} x$

Step 6.3.2.1.3

Multiply $2$ by $1$ .

$x' = 2 x^{\frac{1}{2}} + 2 x^{\frac{1}{2}} x$

Step 6.3.2.1.4

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

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Step 6.3.2.1.4.1

Move $x$ .

$x' = 2 x^{\frac{1}{2}} + 2 (x \cdot x^{\frac{1}{2}})$

Step 6.3.2.1.4.2

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

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Step 6.3.2.1.4.2.1

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

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

Step 6.3.2.1.4.2.2

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

$x' = 2 x^{\frac{1}{2}} + 2 x^{1 + \frac{1}{2}}$

Step 6.3.2.1.4.3

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

$x' = 2 x^{\frac{1}{2}} + 2 x^{\frac{2}{2} + \frac{1}{2}}$

Step 6.3.2.1.4.4

Combine the numerators over the common denominator.

$x' = 2 x^{\frac{1}{2}} + 2 x^{\frac{2 + 1}{2}}$

Step 6.3.2.1.4.5

Add $2$ and $1$ .

$x' = 2 x^{\frac{1}{2}} + 2 x^{\frac{3}{2}}$

Step 7

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

$\frac{d x}{d y} = 2 x^{\frac{1}{2}} + 2 x^{\frac{3}{2}}$