Calculus Examples

$y = x^{2}$

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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

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To apply the Chain Rule, set $u$ as $x$ .

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

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

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

Replace all occurrences of $u$ with $x$ .

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

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

$2 x x'$

Step 4

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

$1 = 2 x x'$

Step 5

Solve for $x'$ .

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Rewrite the equation as $2 x x' = 1$ .

$2 x x' = 1$

Divide each term in $2 x x' = 1$ by $2 x$ and simplify.

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Divide each term in $2 x x' = 1$ by $2 x$ .

$\frac{2 x x'}{2 x} = \frac{1}{2 x}$

Simplify the left side.

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Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 x x'}{2 x} = \frac{1}{2 x}$

Rewrite the expression.

$\frac{x x'}{x} = \frac{1}{2 x}$

Cancel the common factor of $x$ .

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Cancel the common factor.

$\frac{x x'}{x} = \frac{1}{2 x}$

Divide $x'$ by $1$ .

$x' = \frac{1}{2 x}$

Step 6

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

$\frac{d x}{d y} = \frac{1}{2 x}$