Calculus Examples

$u = 2 x$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ 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 $u$ , the derivative of $2 x$ with respect to $u$ is $2 \frac{d}{d u} [x]$ .

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

Step 3.2

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

$2 x'$

Step 4

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

$1 = 2 x'$

Step 5

Solve for $x'$ .

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Step 5.1

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

$2 x' = 1$

Step 5.2

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

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Step 5.2.1

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

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

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.

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

Step 5.2.2.1.2

Divide $x'$ by $1$ .

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

Step 6

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

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