Calculus Examples

$\frac{K x}{(A + x)}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = K x$ and $g (x) = A + x$ .

$\frac{(A + x) \frac{d}{d x} [K x] - K x \frac{d}{d x} [A + x]}{{(A + x)}^{2}}$

Step 2

Differentiate.

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Step 2.1

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

$\frac{(A + x) (K \frac{d}{d x} [x]) - K x \frac{d}{d x} [A + x]}{{(A + x)}^{2}}$

Step 2.2

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

$\frac{(A + x) (K \cdot 1) - K x \frac{d}{d x} [A + x]}{{(A + x)}^{2}}$

Step 2.3

Multiply $K$ by $1$ .

$\frac{(A + x) K - K x \frac{d}{d x} [A + x]}{{(A + x)}^{2}}$

Step 2.4

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

$\frac{(A + x) K - K x (\frac{d}{d x} [A] + \frac{d}{d x} [x])}{{(A + x)}^{2}}$

Step 2.5

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

$\frac{(A + x) K - K x (0 + \frac{d}{d x} [x])}{{(A + x)}^{2}}$

Step 2.6

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

$\frac{(A + x) K - K x (0 + 1)}{{(A + x)}^{2}}$

Step 2.7

Add $0$ and $1$ .

$\frac{(A + x) K - K x \cdot 1}{{(A + x)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{A K + x K - K x \cdot 1}{{(A + x)}^{2}}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Multiply $- 1$ by $1$ .

$\frac{A K + x K - K x}{{(A + x)}^{2}}$

Step 3.2.2

Combine the opposite terms in $A K + x K - K x$ .

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Step 3.2.2.1

Reorder the factors in the terms $x K$ and $- K x$ .

$\frac{A K + K x - K x}{{(A + x)}^{2}}$

Step 3.2.2.2

Subtract $K x$ from $K x$ .

$\frac{A K + 0}{{(A + x)}^{2}}$

Step 3.2.2.3

Add $A K$ and $0$ .

$\frac{A K}{{(A + x)}^{2}}$