Calculus Examples

$x^{2} y + 4 y = 8$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{2} y + 4 y) = \frac{d}{d x} (8)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Move $2$ to the left of $y$ .

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

Step 2.3

Evaluate $\frac{d}{d x} [4 y]$ .

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Step 2.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 y$ with respect to $x$ is $4 \frac{d}{d x} [y]$ .

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

Step 2.3.2

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

$x^{2} y' + 2 y x + 4 y'$

Step 2.4

Reorder terms.

$x^{2} y' + 2 x y + 4 y'$

Step 3

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$0$

Step 4

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

$x^{2} y' + 2 x y + 4 y' = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Subtract $2 x y$ from both sides of the equation.

$x^{2} y' + 4 y' = - 2 x y$

Step 5.2

Factor $y'$ out of $x^{2} y' + 4 y'$ .

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Step 5.2.1

Factor $y'$ out of $x^{2} y'$ .

$y' x^{2} + 4 y' = - 2 x y$

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $x^{2} + 4$ .

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Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

Divide $y'$ by $1$ .

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

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Move the negative in front of the fraction.

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

Step 6

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

$\frac{d y}{d x} = - \frac{2 x y}{x^{2} + 4}$