Calculus Examples

$y \sqrt{x + 1} = 4$

Step 1

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

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

Step 2

Differentiate both sides of the equation.

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

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

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

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 $g (x) = x + 1$ .

Tap for more steps...

Step 3.2.1

To apply the Chain Rule, set $u$ as $x + 1$ .

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

Step 3.2.2

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

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

Step 3.2.3

Replace all occurrences of $u$ with $x + 1$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

Simplify the numerator.

Tap for more steps...

Step 3.6.1

Multiply $- 1$ by $2$ .

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

Step 3.6.2

Subtract $2$ from $1$ .

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

Step 3.7

Combine fractions.

Tap for more steps...

Step 3.7.1

Move the negative in front of the fraction.

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

Step 3.7.2

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

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

Step 3.7.3

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

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

Step 3.7.4

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

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

Step 3.8

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

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

Step 3.9

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

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

Step 3.10

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

$\frac{y}{2 {(x + 1)}^{\frac{1}{2}}} (1 + 0) + {(x + 1)}^{\frac{1}{2}} \frac{d}{d x} [y]$

Step 3.11

Simplify the expression.

Tap for more steps...

Step 3.11.1

Add $1$ and $0$ .

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

Step 3.11.2

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

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

Step 3.12

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

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

Step 3.13

Reorder terms.

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

Step 4

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

$0$

Step 5

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

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

Step 6

Solve for $y'$ .

Tap for more steps...

Step 6.1

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

Tap for more steps...

Step 6.1.1

To write ${(x + 1)}^{\frac{1}{2}} y'$ as a fraction with a common denominator, multiply by $\frac{2 {(x + 1)}^{\frac{1}{2}}}{2 {(x + 1)}^{\frac{1}{2}}}$ .

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

Step 6.1.2

Simplify terms.

Tap for more steps...

Step 6.1.2.1

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

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

Step 6.1.2.2

Combine the numerators over the common denominator.

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

Step 6.1.3

Simplify the numerator.

Tap for more steps...

Step 6.1.3.1

Rewrite using the commutative property of multiplication.

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

Step 6.1.3.2

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

Tap for more steps...

Step 6.1.3.2.1

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

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

Step 6.1.3.2.2

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

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

Step 6.1.3.2.3

Combine the numerators over the common denominator.

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

Step 6.1.3.2.4

Add $1$ and $1$ .

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

Step 6.1.3.2.5

Divide $2$ by $2$ .

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

Step 6.1.3.3

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

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

Step 6.1.3.4

Apply the distributive property.

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

Step 6.1.3.5

Multiply $2$ by $1$ .

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

Step 6.1.3.6

Apply the distributive property.

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

Step 6.2

Set the numerator equal to zero.

$2 x y' + 2 y' + y = 0$

Step 6.3

Solve the equation for $y'$ .

Tap for more steps...

Step 6.3.1

Subtract $y$ from both sides of the equation.

$2 x y' + 2 y' = - y$

Step 6.3.2

Factor $2 y'$ out of $2 x y' + 2 y'$ .

Tap for more steps...

Step 6.3.2.1

Factor $2 y'$ out of $2 x y'$ .

$2 y' x + 2 y' = - y$

Step 6.3.2.2

Factor $2 y'$ out of $2 y'$ .

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

Step 6.3.2.3

Factor $2 y'$ out of $2 y' x + 2 y' \cdot 1$ .

$2 y' (x + 1) = - y$

Step 6.3.3

Divide each term in $2 y' (x + 1) = - y$ by $2 (x + 1)$ and simplify.

Tap for more steps...

Step 6.3.3.1

Divide each term in $2 y' (x + 1) = - y$ by $2 (x + 1)$ .

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

Step 6.3.3.2

Simplify the left side.

Tap for more steps...

Step 6.3.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.3.3.2.1.1

Cancel the common factor.

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

Step 6.3.3.2.1.2

Rewrite the expression.

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

Step 6.3.3.2.2

Cancel the common factor of $x + 1$ .

Tap for more steps...

Step 6.3.3.2.2.1

Cancel the common factor.

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

Step 6.3.3.2.2.2

Divide $y'$ by $1$ .

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

Step 6.3.3.3

Simplify the right side.

Tap for more steps...

Step 6.3.3.3.1

Move the negative in front of the fraction.

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

Step 7

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

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