Calculus Examples

$\sqrt{1 - y^{2}}$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

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

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

Step 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 $g (y) = 1 - y^{2}$ .

Tap for more steps...

Step 1.2.1

To apply the Chain Rule, set $u$ as $1 - y^{2}$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d y} [1 - y^{2}]$

Step 1.2.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d y} [1 - y^{2}]$

Step 1.2.3

Replace all occurrences of $u$ with $1 - y^{2}$ .

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

Combine the numerators over the common denominator.

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

Step 1.6

Simplify the numerator.

Tap for more steps...

Step 1.6.1

Multiply $- 1$ by $2$ .

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

Step 1.6.2

Subtract $2$ from $1$ .

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

Step 1.7

Combine fractions.

Tap for more steps...

Step 1.7.1

Move the negative in front of the fraction.

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

Step 1.7.2

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

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

Step 1.7.3

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

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

Step 1.8

By the Sum Rule, the derivative of $1 - y^{2}$ with respect to $y$ is $\frac{d}{d y} [1] + \frac{d}{d y} [- y^{2}]$ .

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

Step 1.9

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

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

Step 1.10

Add $0$ and $\frac{d}{d y} [- y^{2}]$ .

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

Step 1.11

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

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

Step 1.12

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

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

Step 1.13

Simplify terms.

Tap for more steps...

Step 1.13.1

Multiply $2$ by $- 1$ .

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

Step 1.13.2

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

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

Step 1.13.3

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

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

Step 1.13.4

Factor $2$ out of $- 2 y$ .

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

Step 1.14

Cancel the common factors.

Tap for more steps...

Step 1.14.1

Factor $2$ out of $2 {(1 - y^{2})}^{\frac{1}{2}}$ .

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

Step 1.14.2

Cancel the common factor.

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

Step 1.14.3

Rewrite the expression.

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

Step 1.15

Move the negative in front of the fraction.

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

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = - 1$ and $g (y) = \frac{y}{{(1 - y^{2})}^{\frac{1}{2}}}$ .

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

Step 2.2

Differentiate using the Quotient Rule which states that $\frac{d}{d y} [\frac{f (y)}{g (y)}]$ is $\frac{g (y) \frac{d}{d y} [f (y)] - f (y) \frac{d}{d y} [g (y)]}{{g (y)}^{2}}$ where $f (y) = y$ and $g (y) = {(1 - y^{2})}^{\frac{1}{2}}$ .

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

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.2.1

Cancel the common factor.

Step 2.3.2.2

Rewrite the expression.

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

Step 2.4

Simplify.

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

Step 2.5

Differentiate using the Power Rule.

Tap for more steps...

Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.6

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 $g (y) = 1 - y^{2}$ .

Tap for more steps...

Step 2.6.1

To apply the Chain Rule, set $u$ as $1 - y^{2}$ .

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

Step 2.6.2

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

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

Step 2.6.3

Replace all occurrences of $u$ with $1 - y^{2}$ .

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

Combine the numerators over the common denominator.

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

Step 2.10

Simplify the numerator.

Tap for more steps...

Step 2.10.1

Multiply $- 1$ by $2$ .

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

Step 2.10.2

Subtract $2$ from $1$ .

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

Step 2.11

Combine fractions.

Tap for more steps...

Step 2.11.1

Move the negative in front of the fraction.

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

Step 2.11.2

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

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

Step 2.11.3

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

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

Step 2.11.4

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

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

Step 2.12

By the Sum Rule, the derivative of $1 - y^{2}$ with respect to $y$ is $\frac{d}{d y} [1] + \frac{d}{d y} [- y^{2}]$ .

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

Step 2.13

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

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

Step 2.14

Add $0$ and $\frac{d}{d y} [- y^{2}]$ .

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

Step 2.15

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

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

Step 2.16

Multiply.

Tap for more steps...

Step 2.16.1

Multiply $- 1$ by $- 1$ .

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

Step 2.16.2

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

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

Step 2.17

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

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

Step 2.18

Combine fractions.

Tap for more steps...

Step 2.18.1

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

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

Step 2.18.2

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

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

Step 2.19

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

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

Step 2.20

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

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

Step 2.21

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

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

Step 2.22

Add $1$ and $1$ .

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

Step 2.23

Cancel the common factor.

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

Step 2.24

Rewrite the expression.

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

Step 2.25

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

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

Step 2.26

Combine the numerators over the common denominator.

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

Step 2.27

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

Tap for more steps...

Step 2.27.1

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

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

Step 2.27.2

Combine the numerators over the common denominator.

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

Step 2.27.3

Add $1$ and $1$ .

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

Step 2.27.4

Divide $2$ by $2$ .

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

Step 2.28

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

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

Step 2.29

Add $- y^{2}$ and $y^{2}$ .

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

Step 2.30

Add $1$ and $0$ .

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

Step 2.31

Rewrite $\frac{\frac{1}{{(1 - y^{2})}^{\frac{1}{2}}}}{1 - y^{2}}$ as a product.

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

Step 2.32

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

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

Step 2.33

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

Tap for more steps...

Step 2.33.1

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

Tap for more steps...

Step 2.33.1.1

Raise $1 - y^{2}$ to the power of $1$ .

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

Step 2.33.1.2

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

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

Step 2.33.2

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

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

Step 2.33.3

Combine the numerators over the common denominator.

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

Step 2.33.4

Add $1$ and $2$ .

$- \frac{1}{{(1 - y^{2})}^{\frac{3}{2}}} + \frac{y}{{(1 - y^{2})}^{\frac{1}{2}}} \frac{d}{d y} [- 1]$

Step 2.34

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

$- \frac{1}{{(1 - y^{2})}^{\frac{3}{2}}} + \frac{y}{{(1 - y^{2})}^{\frac{1}{2}}} \cdot 0$

Step 2.35

Simplify the expression.

Tap for more steps...

Step 2.35.1

Multiply $\frac{y}{{(1 - y^{2})}^{\frac{1}{2}}}$ by $0$ .

$- \frac{1}{{(1 - y^{2})}^{\frac{3}{2}}} + 0$

Step 2.35.2

Add $- \frac{1}{{(1 - y^{2})}^{\frac{3}{2}}}$ and $0$ .

$f'' (y) = - \frac{1}{{(1 - y^{2})}^{\frac{3}{2}}}$