Calculus Examples

$x = 2 y$

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

$2 y'$

Step 4

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

$1 = 2 y'$

Step 5

Solve for $y'$ .

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Step 5.1

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

$2 y' = 1$

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Divide $y'$ by $1$ .

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

Step 6

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

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