Calculus Examples

$\frac{d y}{d x} = 2 y$

Differentiate both sides of the equation.

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

Differentiate the left side of the equation.

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Cancel the common factor of $d$ .

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Cancel the common factor.

$\frac{d}{d x} [\frac{d y}{d x}]$

Rewrite the expression.

$\frac{d}{d x} [\frac{y}{x}]$

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 $f (x) = y$ and $g (x) = x$ .

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

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

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

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

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

Multiply $- 1$ by $1$ .

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

Differentiate the right side of the equation.

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

$2 \frac{d}{d x} [y]$

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

$2 y'$

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

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

Solve for $y'$ .

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

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

Simplify the left side.

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

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

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Cancel the common factor.

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

Rewrite the expression.

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

Reorder $x$ and $y'$ .

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

Solve for $y'$ .

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Subtract $2 y' x^{2}$ from both sides of the equation.

$y' x - y - 2 y' x^{2} = 0$

Add $y$ to both sides of the equation.

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

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

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

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

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

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

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

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

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

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

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

Simplify the left side.

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Cancel the common factor of $x$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $1 - 2 x$ .

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Cancel the common factor.

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

Divide $y'$ by $1$ .

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

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

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