Calculus Examples

$y^{2} (y^{2} - 4) = x^{2} (x^{2} - 5)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y^{2} (y^{2} - 4)) = \frac{d}{d x} (x^{2} (x^{2} - 5))$

Step 2

Differentiate the left side of the equation.

Tap for more steps...

Step 2.1

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 $f (x) = y^{2}$ and $g (x) = y^{2} - 4$ .

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

Step 2.2

By the Sum Rule, the derivative of $y^{2} - 4$ with respect to $x$ is $\frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 4]$ .

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

Step 2.3

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 2.3.1

To apply the Chain Rule, set $u_{1}$ as $y$ .

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

Step 2.3.2

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

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

Step 2.3.3

Replace all occurrences of $u_{1}$ with $y$ .

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

Step 2.4

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

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

Step 2.5

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

$y^{2} (2 y y' + 0) + (y^{2} - 4) \frac{d}{d x} [y^{2}]$

Step 2.6

Add $2 y y'$ and $0$ .

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

Step 2.7

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

Tap for more steps...

Step 2.7.1

Move $y$ .

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

Step 2.7.2

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

Tap for more steps...

Step 2.7.2.1

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

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

Step 2.7.2.2

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

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

Step 2.7.3

Add $1$ and $2$ .

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

Step 2.8

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 2.8.1

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

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

Step 2.8.2

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

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

Step 2.8.3

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

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

Step 2.9

Move $2$ to the left of $y^{2} - 4$ .

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

Step 2.10

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

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

Step 2.11

Simplify.

Tap for more steps...

Step 2.11.1

Apply the distributive property.

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

Step 2.11.2

Apply the distributive property.

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

Step 2.11.3

Apply the distributive property.

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

Step 2.11.4

Combine terms.

Tap for more steps...

Step 2.11.4.1

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

$y^{3} (2 y') + 2 (y^{1} y^{2}) y' + 2 \cdot - 4 y y'$

Step 2.11.4.2

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

$y^{3} (2 y') + 2 y^{1 + 2} y' + 2 \cdot - 4 y y'$

Step 2.11.4.3

Add $1$ and $2$ .

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

Step 2.11.4.4

Multiply $2$ by $- 4$ .

$y^{3} (2 y') + 2 y^{3} y' - 8 y y'$

Step 2.11.4.5

Add $y^{3} (2 y')$ and $2 y^{3} y'$ .

Tap for more steps...

Step 2.11.4.5.1

Reorder $y^{3}$ and $2$ .

$2 \cdot y^{3} y' + 2 y^{3} y' - 8 y y'$

Step 2.11.4.5.2

Add $2 \cdot y^{3} y'$ and $2 y^{3} y'$ .

$4 y^{3} y' - 8 y y'$

Step 3

Differentiate the right 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 $f (x) = x^{2}$ and $g (x) = x^{2} - 5$ .

$x^{2} \frac{d}{d x} [x^{2} - 5] + (x^{2} - 5) \frac{d}{d x} [x^{2}]$

Step 3.2

Differentiate.

Tap for more steps...

Step 3.2.1

By the Sum Rule, the derivative of $x^{2} - 5$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5]$ .

$x^{2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5]) + (x^{2} - 5) \frac{d}{d x} [x^{2}]$

Step 3.2.2

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

$x^{2} (2 x + \frac{d}{d x} [- 5]) + (x^{2} - 5) \frac{d}{d x} [x^{2}]$

Step 3.2.3

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$x^{2} (2 x + 0) + (x^{2} - 5) \frac{d}{d x} [x^{2}]$

Step 3.2.4

Add $2 x$ and $0$ .

$x^{2} (2 x) + (x^{2} - 5) \frac{d}{d x} [x^{2}]$

Step 3.3

Multiply $x^{2}$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.3.1

Move $x$ .

$x \cdot x^{2} \cdot 2 + (x^{2} - 5) \frac{d}{d x} [x^{2}]$

Step 3.3.2

Multiply $x$ by $x^{2}$ .

Tap for more steps...

Step 3.3.2.1

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

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

Step 3.3.2.2

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

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

Step 3.3.3

Add $1$ and $2$ .

$x^{3} \cdot 2 + (x^{2} - 5) \frac{d}{d x} [x^{2}]$

Step 3.4

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

$2 \cdot x^{3} + (x^{2} - 5) \frac{d}{d x} [x^{2}]$

Step 3.5

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

$2 x^{3} + (x^{2} - 5) (2 x)$

Step 3.6

Move $2$ to the left of $x^{2} - 5$ .

$2 x^{3} + 2 \cdot (x^{2} - 5) x$

Step 3.7

Simplify.

Tap for more steps...

Step 3.7.1

Apply the distributive property.

$2 x^{3} + (2 x^{2} + 2 \cdot - 5) x$

Step 3.7.2

Apply the distributive property.

$2 x^{3} + 2 x^{2} x + 2 \cdot - 5 x$

Step 3.7.3

Combine terms.

Tap for more steps...

Step 3.7.3.1

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

$2 x^{3} + 2 (x^{1} x^{2}) + 2 \cdot - 5 x$

Step 3.7.3.2

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

$2 x^{3} + 2 x^{1 + 2} + 2 \cdot - 5 x$

Step 3.7.3.3

Add $1$ and $2$ .

$2 x^{3} + 2 x^{3} + 2 \cdot - 5 x$

Step 3.7.3.4

Multiply $2$ by $- 5$ .

$2 x^{3} + 2 x^{3} - 10 x$

Step 3.7.3.5

Add $2 x^{3}$ and $2 x^{3}$ .

$4 x^{3} - 10 x$

Step 4

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

$4 y^{3} y' - 8 y y' = 4 x^{3} - 10 x$

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

Factor $4 y y'$ out of $4 y^{3} y' - 8 y y'$ .

Tap for more steps...

Step 5.1.1

Factor $4 y y'$ out of $4 y^{3} y'$ .

$4 y y' y^{2} - 8 y y' = 4 x^{3} - 10 x$

Step 5.1.2

Factor $4 y y'$ out of $- 8 y y'$ .

$4 y y' y^{2} + 4 y y' \cdot - 2 = 4 x^{3} - 10 x$

Step 5.1.3

Factor $4 y y'$ out of $4 y y' y^{2} + 4 y y' \cdot - 2$ .

$4 y y' (y^{2} - 2) = 4 x^{3} - 10 x$

Step 5.2

Divide each term in $4 y y' (y^{2} - 2) = 4 x^{3} - 10 x$ by $4 y (y^{2} - 2)$ and simplify.

Tap for more steps...

Step 5.2.1

Divide each term in $4 y y' (y^{2} - 2) = 4 x^{3} - 10 x$ by $4 y (y^{2} - 2)$ .

$\frac{4 y y' (y^{2} - 2)}{4 y (y^{2} - 2)} = \frac{4 x^{3}}{4 y (y^{2} - 2)} + \frac{- 10 x}{4 y (y^{2} - 2)}$

Step 5.2.2

Simplify the left side.

Tap for more steps...

Step 5.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 5.2.2.1.1

Cancel the common factor.

$\frac{4 y y' (y^{2} - 2)}{4 y (y^{2} - 2)} = \frac{4 x^{3}}{4 y (y^{2} - 2)} + \frac{- 10 x}{4 y (y^{2} - 2)}$

Step 5.2.2.1.2

Rewrite the expression.

$\frac{y y' (y^{2} - 2)}{y (y^{2} - 2)} = \frac{4 x^{3}}{4 y (y^{2} - 2)} + \frac{- 10 x}{4 y (y^{2} - 2)}$

Step 5.2.2.2

Cancel the common factor of $y$ .

Tap for more steps...

Step 5.2.2.2.1

Cancel the common factor.

$\frac{y y' (y^{2} - 2)}{y (y^{2} - 2)} = \frac{4 x^{3}}{4 y (y^{2} - 2)} + \frac{- 10 x}{4 y (y^{2} - 2)}$

Step 5.2.2.2.2

Rewrite the expression.

$\frac{y' (y^{2} - 2)}{y^{2} - 2} = \frac{4 x^{3}}{4 y (y^{2} - 2)} + \frac{- 10 x}{4 y (y^{2} - 2)}$

Step 5.2.2.3

Cancel the common factor of $y^{2} - 2$ .

Tap for more steps...

Step 5.2.2.3.1

Cancel the common factor.

$\frac{y' (y^{2} - 2)}{y^{2} - 2} = \frac{4 x^{3}}{4 y (y^{2} - 2)} + \frac{- 10 x}{4 y (y^{2} - 2)}$

Step 5.2.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{4 x^{3}}{4 y (y^{2} - 2)} + \frac{- 10 x}{4 y (y^{2} - 2)}$

Step 5.2.3

Simplify the right side.

Tap for more steps...

Step 5.2.3.1

Simplify each term.

Tap for more steps...

Step 5.2.3.1.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 5.2.3.1.1.1

Cancel the common factor.

$y' = \frac{4 x^{3}}{4 y (y^{2} - 2)} + \frac{- 10 x}{4 y (y^{2} - 2)}$

Step 5.2.3.1.1.2

Rewrite the expression.

$y' = \frac{x^{3}}{y (y^{2} - 2)} + \frac{- 10 x}{4 y (y^{2} - 2)}$

Step 5.2.3.1.2

Cancel the common factor of $- 10$ and $4$ .

Tap for more steps...

Step 5.2.3.1.2.1

Factor $2$ out of $- 10 x$ .

$y' = \frac{x^{3}}{y (y^{2} - 2)} + \frac{2 (- 5 x)}{4 y (y^{2} - 2)}$

Step 5.2.3.1.2.2

Cancel the common factors.

Tap for more steps...

Step 5.2.3.1.2.2.1

Factor $2$ out of $4 y (y^{2} - 2)$ .

$y' = \frac{x^{3}}{y (y^{2} - 2)} + \frac{2 (- 5 x)}{2 (2 y (y^{2} - 2))}$

Step 5.2.3.1.2.2.2

Cancel the common factor.

$y' = \frac{x^{3}}{y (y^{2} - 2)} + \frac{2 (- 5 x)}{2 (2 y (y^{2} - 2))}$

Step 5.2.3.1.2.2.3

Rewrite the expression.

$y' = \frac{x^{3}}{y (y^{2} - 2)} + \frac{- 5 x}{2 y (y^{2} - 2)}$

Step 5.2.3.1.3

Move the negative in front of the fraction.

$y' = \frac{x^{3}}{y (y^{2} - 2)} - \frac{5 x}{2 y (y^{2} - 2)}$

Step 6

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

$\frac{d y}{d x} = \frac{x^{3}}{y (y^{2} - 2)} - \frac{5 x}{2 y (y^{2} - 2)}$