Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

Tap for more steps...

Step 2.1

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

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

Step 2.2

Evaluate $\frac{d}{d x} [\frac{1}{x^{2}}]$ .

Tap for more steps...

Step 2.2.1

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

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

Step 2.2.2

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

Tap for more steps...

Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

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

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

Step 2.2.3

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

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

Step 2.2.4

Multiply the exponents in ${(x^{2})}^{- 2}$ .

Tap for more steps...

Step 2.2.4.1

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

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

Step 2.2.4.2

Multiply $2$ by $- 2$ .

$- x^{- 4} (2 x) + \frac{d}{d x} [\frac{1}{y^{2}}]$

Step 2.2.5

Multiply $2$ by $- 1$ .

$- 2 x^{- 4} x + \frac{d}{d x} [\frac{1}{y^{2}}]$

Step 2.2.6

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

$- 2 (x^{1} x^{- 4}) + \frac{d}{d x} [\frac{1}{y^{2}}]$

Step 2.2.7

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

$- 2 x^{1 - 4} + \frac{d}{d x} [\frac{1}{y^{2}}]$

Step 2.2.8

Subtract $4$ from $1$ .

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

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

Tap for more steps...

Step 2.3.3.1

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

$- 2 x^{- 3} + \frac{y^{2} \cdot 0 - 1 \cdot 1 (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [y])}{{(y^{2})}^{2}}$

Step 2.3.3.2

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

$- 2 x^{- 3} + \frac{y^{2} \cdot 0 - 1 \cdot 1 (2 u_{2} \frac{d}{d x} [y])}{{(y^{2})}^{2}}$

Step 2.3.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Multiply $y^{2}$ by $0$ .

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

Step 2.3.6

Multiply $- 1$ by $1$ .

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

Step 2.3.7

Multiply $2$ by $- 1$ .

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

Step 2.3.8

Subtract $2 y y'$ from $0$ .

$- 2 x^{- 3} + \frac{- 2 y y'}{{(y^{2})}^{2}}$

Step 2.3.9

Multiply the exponents in ${(y^{2})}^{2}$ .

Tap for more steps...

Step 2.3.9.1

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

$- 2 x^{- 3} + \frac{- 2 y y'}{y^{2 \cdot 2}}$

Step 2.3.9.2

Multiply $2$ by $2$ .

$- 2 x^{- 3} + \frac{- 2 y y'}{y^{4}}$

Step 2.3.10

Cancel the common factor of $y$ and $y^{4}$ .

Tap for more steps...

Step 2.3.10.1

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

$- 2 x^{- 3} + \frac{y (- 2 y')}{y^{4}}$

Step 2.3.10.2

Cancel the common factors.

Tap for more steps...

Step 2.3.10.2.1

Factor $y$ out of $y^{4}$ .

$- 2 x^{- 3} + \frac{y (- 2 y')}{y \cdot y^{3}}$

Step 2.3.10.2.2

Cancel the common factor.

$- 2 x^{- 3} + \frac{y (- 2 y')}{y \cdot y^{3}}$

Step 2.3.10.2.3

Rewrite the expression.

$- 2 x^{- 3} + \frac{- 2 y'}{y^{3}}$

Step 2.3.11

Move the negative in front of the fraction.

$- 2 x^{- 3} - \frac{2 y'}{y^{3}}$

Step 2.4

Simplify.

Tap for more steps...

Step 2.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 2 \frac{1}{x^{3}} - \frac{2 y'}{y^{3}}$

Step 2.4.2

Combine terms.

Tap for more steps...

Step 2.4.2.1

Combine $- 2$ and $\frac{1}{x^{3}}$ .

$\frac{- 2}{x^{3}} - \frac{2 y'}{y^{3}}$

Step 2.4.2.2

Move the negative in front of the fraction.

$- \frac{2}{x^{3}} - \frac{2 y'}{y^{3}}$

Step 3

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

$0$

Step 4

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

$- \frac{2}{x^{3}} - \frac{2 y'}{y^{3}} = 0$

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

Add $\frac{2}{x^{3}}$ to both sides of the equation.

$- \frac{2 y'}{y^{3}} = \frac{2}{x^{3}}$

Step 5.2

Divide each term in $- \frac{2 y'}{y^{3}} = \frac{2}{x^{3}}$ by $- 1$ and simplify.

Tap for more steps...

Step 5.2.1

Divide each term in $- \frac{2 y'}{y^{3}} = \frac{2}{x^{3}}$ by $- 1$ .

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

Step 5.2.2

Simplify the left side.

Tap for more steps...

Step 5.2.2.1

Dividing two negative values results in a positive value.

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

Step 5.2.2.2

Divide $\frac{2 y'}{y^{3}}$ by $1$ .

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

Step 5.2.3

Simplify the right side.

Tap for more steps...

Step 5.2.3.1

Move the negative one from the denominator of $\frac{\frac{2}{x^{3}}}{- 1}$ .

$\frac{2 y'}{y^{3}} = - 1 \cdot \frac{2}{x^{3}}$

Step 5.2.3.2

Rewrite $- 1 \cdot \frac{2}{x^{3}}$ as $- \frac{2}{x^{3}}$ .

$\frac{2 y'}{y^{3}} = - \frac{2}{x^{3}}$

Step 5.3

Multiply both sides by $y^{3}$ .

$\frac{2 y'}{y^{3}} y^{3} = - \frac{2}{x^{3}} y^{3}$

Step 5.4

Simplify.

Tap for more steps...

Step 5.4.1

Simplify the left side.

Tap for more steps...

Step 5.4.1.1

Cancel the common factor of $y^{3}$ .

Tap for more steps...

Step 5.4.1.1.1

Cancel the common factor.

$\frac{2 y'}{y^{3}} y^{3} = - \frac{2}{x^{3}} y^{3}$

Step 5.4.1.1.2

Rewrite the expression.

$2 y' = - \frac{2}{x^{3}} y^{3}$

Step 5.4.2

Simplify the right side.

Tap for more steps...

Step 5.4.2.1

Simplify $- \frac{2}{x^{3}} y^{3}$ .

Tap for more steps...

Step 5.4.2.1.1

Combine $y^{3}$ and $\frac{2}{x^{3}}$ .

$2 y' = - \frac{y^{3} \cdot 2}{x^{3}}$

Step 5.4.2.1.2

Move $2$ to the left of $y^{3}$ .

$2 y' = - \frac{2 y^{3}}{x^{3}}$

Step 5.5

Divide each term in $2 y' = - \frac{2 y^{3}}{x^{3}}$ by $2$ and simplify.

Tap for more steps...

Step 5.5.1

Divide each term in $2 y' = - \frac{2 y^{3}}{x^{3}}$ by $2$ .

$\frac{2 y'}{2} = \frac{- \frac{2 y^{3}}{x^{3}}}{2}$

Step 5.5.2

Simplify the left side.

Tap for more steps...

Step 5.5.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.5.2.1.1

Cancel the common factor.

$\frac{2 y'}{2} = \frac{- \frac{2 y^{3}}{x^{3}}}{2}$

Step 5.5.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- \frac{2 y^{3}}{x^{3}}}{2}$

Step 5.5.3

Simplify the right side.

Tap for more steps...

Step 5.5.3.1

Multiply the numerator by the reciprocal of the denominator.

$y' = - \frac{2 y^{3}}{x^{3}} \cdot \frac{1}{2}$

Step 5.5.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.5.3.2.1

Move the leading negative in $- \frac{2 y^{3}}{x^{3}}$ into the numerator.

$y' = \frac{- 2 y^{3}}{x^{3}} \cdot \frac{1}{2}$

Step 5.5.3.2.2

Factor $2$ out of $- 2 y^{3}$ .

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

Step 5.5.3.2.3

Cancel the common factor.

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

Step 5.5.3.2.4

Rewrite the expression.

$y' = \frac{- y^{3}}{x^{3}}$

Step 5.5.3.3

Move the negative in front of the fraction.

$y' = - \frac{y^{3}}{x^{3}}$

Step 6

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

$\frac{d y}{d x} = - \frac{y^{3}}{x^{3}}$