Calculus Examples

$5.5 = 14 s - 14 x$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (5.5) = \frac{d}{d x} (14 s - 14 x)$

Step 2

Since $5.5$ is constant with respect to $x$ , the derivative of $5.5$ with respect to $x$ is $0$ .

$0$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $14 s - 14 x$ with respect to $x$ is $\frac{d}{d x} [14 s] + \frac{d}{d x} [- 14 x]$ .

$\frac{d}{d x} [14 s] + \frac{d}{d x} [- 14 x]$

Step 3.2

Evaluate $\frac{d}{d x} [14 s]$ .

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Step 3.2.1

Since $14$ is constant with respect to $x$ , the derivative of $14 s$ with respect to $x$ is $14 \frac{d}{d x} [s]$ .

$14 \frac{d}{d x} [s] + \frac{d}{d x} [- 14 x]$

Step 3.2.2

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

$14 s' + \frac{d}{d x} [- 14 x]$

Step 3.3

Evaluate $\frac{d}{d x} [- 14 x]$ .

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Step 3.3.1

Since $- 14$ is constant with respect to $x$ , the derivative of $- 14 x$ with respect to $x$ is $- 14 \frac{d}{d x} [x]$ .

$14 s' - 14 \frac{d}{d x} [x]$

Step 3.3.2

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

$14 s' - 14 \cdot 1$

Step 3.3.3

Multiply $- 14$ by $1$ .

$14 s' - 14$

Step 4

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

$0 = 14 s' - 14$

Step 5

Solve for $s'$ .

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Step 5.1

Rewrite the equation as $14 s' - 14 = 0$ .

$14 s' - 14 = 0$

Step 5.2

Add $14$ to both sides of the equation.

$14 s' = 14$

Step 5.3

Divide each term in $14 s' = 14$ by $14$ and simplify.

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Step 5.3.1

Divide each term in $14 s' = 14$ by $14$ .

$\frac{14 s'}{14} = \frac{14}{14}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $14$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{14 s'}{14} = \frac{14}{14}$

Step 5.3.2.1.2

Divide $s'$ by $1$ .

$s' = \frac{14}{14}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Divide $14$ by $14$ .

$s' = 1$

Step 6

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

$\frac{d s}{d x} = 1$