Find dx/dy y=10 square root of x

$y = 10 \sqrt{x}$

Step 1

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

$y = 10 x^{\frac{1}{2}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d y} (y) = \frac{d}{d y} (10 x^{\frac{1}{2}})$

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.

Since $10$ is constant with respect to $y$ , the derivative of $10 x^{\frac{1}{2}}$ with respect to $y$ is $10 \frac{d}{d y} [x^{\frac{1}{2}}]$ .

$10 \frac{d}{d y} [x^{\frac{1}{2}}]$

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 $u$ as $x$ .

$10 (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d y} [x])$

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

$10 (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d y} [x])$

Replace all occurrences of $u$ with $x$ .

$10 (\frac{1}{2} x^{\frac{1}{2} - 1} \frac{d}{d y} [x])$

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

$10 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d y} [x])$

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

$10 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d y} [x])$

Combine the numerators over the common denominator.

$10 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d y} [x])$

Simplify the numerator.

Multiply $- 1$ by $2$ .

$10 (\frac{1}{2} x^{\frac{1 - 2}{2}} \frac{d}{d y} [x])$

Subtract $2$ from $1$ .

$10 (\frac{1}{2} x^{\frac{- 1}{2}} \frac{d}{d y} [x])$

Move the negative in front of the fraction.

$10 (\frac{1}{2} x^{- \frac{1}{2}} \frac{d}{d y} [x])$

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

$10 (\frac{x^{- \frac{1}{2}}}{2} \frac{d}{d y} [x])$

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

$10 (\frac{1}{2 x^{\frac{1}{2}}} \frac{d}{d y} [x])$

Combine $\frac{1}{2 x^{\frac{1}{2}}}$ and $10$ .

$\frac{10}{2 x^{\frac{1}{2}}} \frac{d}{d y} [x]$

Factor $2$ out of $10$ .

$\frac{2 \cdot 5}{2 x^{\frac{1}{2}}} \frac{d}{d y} [x]$

Cancel the common factors.

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

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

Cancel the common factor.

$\frac{2 \cdot 5}{2 x^{\frac{1}{2}}} \frac{d}{d y} [x]$

Rewrite the expression.

$\frac{5}{x^{\frac{1}{2}}} \frac{d}{d y} [x]$

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

$\frac{5}{x^{\frac{1}{2}}} x'$

Combine $\frac{5}{x^{\frac{1}{2}}}$ and $x'$ .

$\frac{5 x'}{x^{\frac{1}{2}}}$

Step 5

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

$1 = \frac{5 x'}{x^{\frac{1}{2}}}$

Step 6

Solve for $x'$ .

Rewrite the equation as $\frac{5 x'}{x^{\frac{1}{2}}} = 1$ .

$\frac{5 x'}{x^{\frac{1}{2}}} = 1$

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

$\frac{5 x'}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 1 x^{\frac{1}{2}}$

Simplify.

Simplify the left side.

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

Cancel the common factor.

$\frac{5 x'}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 1 x^{\frac{1}{2}}$

Rewrite the expression.

$5 x' = 1 x^{\frac{1}{2}}$

Simplify the right side.

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

$5 x' = x^{\frac{1}{2}}$

Divide each term in $5 x' = x^{\frac{1}{2}}$ by $5$ and simplify.

Divide each term in $5 x' = x^{\frac{1}{2}}$ by $5$ .

$\frac{5 x'}{5} = \frac{x^{\frac{1}{2}}}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x'}{5} = \frac{x^{\frac{1}{2}}}{5}$

Divide $x'$ by $1$ .

$x' = \frac{x^{\frac{1}{2}}}{5}$

Step 7

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

$\frac{d x}{d y} = \frac{x^{\frac{1}{2}}}{5}$

Find dx/dy y=10 square root of x

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