Calculus Examples

$x = 8 u v$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d u} (x) = \frac{d}{d u} (8 u x')$

Step 2

The derivative of $x$ with respect to $u$ is $x'$ .

$x'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $8 x'$ is constant with respect to $u$ , the derivative of $8 u x'$ with respect to $u$ is $8 x' \frac{d}{d u} [u]$ .

$8 x' \frac{d}{d u} [u]$

Step 3.2

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

$8 x' \cdot 1$

Step 3.3

Multiply $8$ by $1$ .

$8 x'$

Step 4

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

$x' = 8 x'$

Step 5

Solve for $x'$ .

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Step 5.1

Move all terms containing $x'$ to the left side of the equation.

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Step 5.1.1

Subtract $8 x'$ from both sides of the equation.

$x' - 8 x' = 0$

Step 5.1.2

Subtract $8 x'$ from $x'$ .

$- 7 x' = 0$

Step 5.2

Divide each term in $- 7 x' = 0$ by $- 7$ and simplify.

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Step 5.2.1

Divide each term in $- 7 x' = 0$ by $- 7$ .

$\frac{- 7 x'}{- 7} = \frac{0}{- 7}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 7$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 7 x'}{- 7} = \frac{0}{- 7}$

Step 5.2.2.1.2

Divide $x'$ by $1$ .

$x' = \frac{0}{- 7}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $0$ by $- 7$ .

$x' = 0$

Step 6

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

$\frac{d x}{d u} = 0$