Find dx/dy y=arctan( square root of x)

$y = \arctan (\sqrt{x})$

Step 1

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

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

Step 2

Differentiate both sides of the equation.

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

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.

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

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

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 $g (y) = x$ .

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 1$ by $2$ .

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

Subtract $2$ from $1$ .

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

Move the negative in front of the fraction.

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

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

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

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

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

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

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

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

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

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 $x'$ .

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

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

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

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

Simplify.

Simplify the left side.

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

Rewrite using the commutative property of multiplication.

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

Cancel the common factor of $2$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the right side.

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

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

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

Apply the distributive property.

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

Multiply $2$ by $1$ .

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

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

Move $x$ .

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Add $2$ and $1$ .

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

Step 7

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

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

Find dx/dy y=arctan( square root of x)

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