Calculus Examples

$S + (x) a x = f (x) + c$

Step 1

Multiply $x$ by $a$ .

$S + x a x = f (x) + c$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (S + x a x) = \frac{d}{d x} (f (x) + c)$

Step 3

Differentiate the left side of the equation.

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Step 3.1

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

$\frac{d}{d x} [S] + \frac{d}{d x} [x a x]$

Step 3.2

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

$S' + \frac{d}{d x} [x a x]$

Step 3.3

Evaluate $\frac{d}{d x} [x a x]$ .

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Step 3.3.1

Raise $x$ to the power of $1$ .

$S' + \frac{d}{d x} [a (x^{1} x)]$

Step 3.3.2

Raise $x$ to the power of $1$ .

$S' + \frac{d}{d x} [a (x^{1} x^{1})]$

Step 3.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$S' + \frac{d}{d x} [a x^{1 + 1}]$

Step 3.3.4

Add $1$ and $1$ .

$S' + \frac{d}{d x} [a x^{2}]$

Step 3.3.5

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

$S' + a \frac{d}{d x} [x^{2}]$

Step 3.3.6

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

$S' + a (2 x)$

Step 3.3.7

Move $2$ to the left of $a$ .

$S' + 2 a x$

Step 3.4

Reorder terms.

$2 a x + S'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

By the Sum Rule, the derivative of $f (x) + c$ with respect to $x$ is $\frac{d}{d x} [f (x)] + \frac{d}{d x} [c]$ .

$\frac{d}{d x} [f (x)] + \frac{d}{d x} [c]$

Step 4.2

Differentiate using the function rule which states that the derivative of $f (x)$ is $\frac{d f}{d x}$ .

$\frac{d f}{d x} + \frac{d}{d x} [c]$

Step 4.3

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

$\frac{d f}{d x} + 0$

Step 4.4

Add $\frac{d f}{d x}$ and $0$ .

$\frac{d f}{d x}$

Step 5

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

$2 a x + S' = \frac{d f}{d x}$

Step 6

Solve for $S'$ .

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Step 6.1

Cancel the common factor of $d$ .

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Step 6.1.1

Cancel the common factor.

$2 a x + S' = \frac{d f}{d x}$

Step 6.1.2

Rewrite the expression.

$2 a x + S' = \frac{f}{x}$

Step 6.2

Subtract $2 a x$ from both sides of the equation.

$S' = \frac{f}{x} - 2 a x$

Step 7

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

$\frac{d S}{d x} = \frac{f}{x} - 2 a x$