Calculus Examples

$x \sqrt{y} = 1$

Step 1

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

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

Step 2

Differentiate both sides of the equation.

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

Step 3

Differentiate the left side of the equation.

Tap for more steps...

Step 3.1

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

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

Step 3.2

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 $g (x) = y$ .

Tap for more steps...

Step 3.2.1

To apply the Chain Rule, set $u$ as $y$ .

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

Simplify the numerator.

Tap for more steps...

Step 3.6.1

Multiply $- 1$ by $2$ .

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

Step 3.6.2

Subtract $2$ from $1$ .

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

Step 3.7

Move the negative in front of the fraction.

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

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

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

Step 3.11

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

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

Step 3.12

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

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

Step 3.13

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

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

Step 3.14

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

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

Step 4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0$

Step 5

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

$\frac{x y'}{2 y^{\frac{1}{2}}} + y^{\frac{1}{2}} = 0$

Step 6

Solve for $y'$ .

Tap for more steps...

Step 6.1

Find a common factor $y^{\frac{1}{2}}$ that is present in each term.

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

Step 6.2

Substitute $u$ for $y^{\frac{1}{2}}$ .

$\frac{1}{2 {(x {y'}^{\frac{1}{2}})}^{2}} + u = 0$

Step 6.3

Solve for $u$ .

Tap for more steps...

Step 6.3.1

Simplify the denominator.

Tap for more steps...

Step 6.3.1.1

Apply the product rule to $x {y'}^{\frac{1}{2}}$ .

$\frac{1}{2 x^{2} {({y'}^{\frac{1}{2}})}^{2}} + u = 0$

Step 6.3.1.2

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

Tap for more steps...

Step 6.3.1.2.1

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

$\frac{1}{2 x^{2} {y'}^{\frac{1}{2} \cdot 2}} + u = 0$

Step 6.3.1.2.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.3.1.2.2.1

Cancel the common factor.

$\frac{1}{2 x^{2} {y'}^{\frac{1}{2} \cdot 2}} + u = 0$

Step 6.3.1.2.2.2

Rewrite the expression.

$\frac{1}{2 x^{2} {y'}^{1}} + u = 0$

Step 6.3.1.3

Simplify.

$\frac{1}{2 x^{2} y'} + u = 0$

Step 6.3.2

Subtract $\frac{1}{2 x^{2} y'}$ from both sides of the equation.

$u = - \frac{1}{2 x^{2} y'}$

Step 6.4

Substitute $y'$ for $u$ .

$y^{\frac{1}{2}} = - \frac{1}{2 x^{2} y'}$

Step 6.5

Subtract $y^{\frac{1}{2}}$ from both sides of the equation.

$\frac{x y'}{2 y^{\frac{1}{2}}} = - y^{\frac{1}{2}}$

Step 6.6

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

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

Step 6.7

Simplify.

Tap for more steps...

Step 6.7.1

Simplify the left side.

Tap for more steps...

Step 6.7.1.1

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

Tap for more steps...

Step 6.7.1.1.1

Rewrite using the commutative property of multiplication.

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

Step 6.7.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.7.1.1.2.1

Cancel the common factor.

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

Step 6.7.1.1.2.2

Rewrite the expression.

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

Step 6.7.1.1.3

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

Tap for more steps...

Step 6.7.1.1.3.1

Cancel the common factor.

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

Step 6.7.1.1.3.2

Rewrite the expression.

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

Step 6.7.2

Simplify the right side.

Tap for more steps...

Step 6.7.2.1

Simplify $- y^{\frac{1}{2}} (2 y^{\frac{1}{2}})$ .

Tap for more steps...

Step 6.7.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 6.7.2.1.2

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

Tap for more steps...

Step 6.7.2.1.2.1

Move $y^{\frac{1}{2}}$ .

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

Step 6.7.2.1.2.2

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

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

Step 6.7.2.1.2.3

Combine the numerators over the common denominator.

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

Step 6.7.2.1.2.4

Add $1$ and $1$ .

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

Step 6.7.2.1.2.5

Divide $2$ by $2$ .

$x y' = - 1 \cdot 2 y^{1}$

Step 6.7.2.1.3

Simplify $- 1 \cdot 2 y^{1}$ .

$x y' = - 1 \cdot 2 y$

Step 6.7.2.1.4

Multiply $- 1$ by $2$ .

$x y' = - 2 y$

Step 6.8

Divide each term in $x y' = - 2 y$ by $x$ and simplify.

Tap for more steps...

Step 6.8.1

Divide each term in $x y' = - 2 y$ by $x$ .

$\frac{x y'}{x} = \frac{- 2 y}{x}$

Step 6.8.2

Simplify the left side.

Tap for more steps...

Step 6.8.2.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 6.8.2.1.1

Cancel the common factor.

$\frac{x y'}{x} = \frac{- 2 y}{x}$

Step 6.8.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- 2 y}{x}$

Step 6.8.3

Simplify the right side.

Tap for more steps...

Step 6.8.3.1

Move the negative in front of the fraction.

$y' = - \frac{2 y}{x}$

Step 7

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

$\frac{d y}{d x} = - \frac{2 y}{x}$