## The Quotient Rule

The& **quotient rule** is used to differentiate functions that are being divided. If you have a function g(x) (top function) divided by h(x) (bottom function) then the quotient rule is:

It looks ugly, but it’s nothing more complicated than following a few steps (which are **exactly the same** for each quotient). You might also notice that the numerator in the quotient rule is the same as the product rule with one slight difference—the addition sign has been replaced with the subtraction sign.

Watch the video or read on below:

## Quotient Rule: Examples

**Example Problem #1: ** Differentiate the following function:

**y = 2 / (x + 1)**

**Solution**:

**Note**: I’m using D as shorthand for derivative here instead of writing g'(x) or f'(x):

**Identify g(x) and h(x)**. The top function (2) is g(x) and the bottom function (x + 1) is f(x).**Plug your functions (from Step 1) into the formula:**

y’ = D{2} (x + 1) – D {x + 1} (2) / (x + 1)^{2}**Work out your derivatives**. For example, the derivative of 2 is 0. y’ = (0)(x + 1) – (1)(2) / (x + 1)^{2}**Simplify**: y’ = -2 (x + 1)^{2}

When working with the quotient rule, **always start with the bottom function**, ending with the bottom function squared.

More examples for the Quotient Rule:

- How to Differentiate (2x + 1) / (x – 3)
- How to Differentiate tan(x)
- How to Differentiate 2
^{x}/ 2^{x}x

## How to Differentiate (2x + 1) / (x – 3)

Step 1: Name the top term (the denominator) f(x) and the bottom term (the numerator) g(x). This gives you two new functions:

- f(x) = 2x + 1
- g(x) = x – 3

Step 2: **Place your functions** f(x) and g(x) into the quotient rule. I’ll use d/dx here to indicate a derivative.

f'(x) = (x – 3) d/dx [2x + 1] – (2x + 1) d/dx[x – 3] / [x-3]^{2}

Step 3:Differentiate the indicated functions in Step 2. In this example, those functions are [2x + 1] and [x + 3].

f'(x) = (x – 3)(2)-(2x + 1)(1) / (x – 3)^{2}

Step 4:**Use algebra** to simplify where possible. The solution is 7/(x – 3)^{2}.

## How to Differentiate tan(x)

The **quotient rule** can be used to differentiate tan(x), because of a basic quotient identity, taken from trigonometry: tan(x) = sin(x) / cos(x).

Step 1: **Name** the top term f(x) and the bottom term g(x). Using our quotient trigonometric identity tan(x) = sinx(x) / cos(s), then:

- f(x) = sin(x)
- g(x) = cos(x)

Step 2: **Place your functions** f(x) and g(x) into the **quotient rule**. The term d/dx here indicates a derivative.

f'(x) = cos(x) d/dx[sin(x)] – sin(x) d/dx[cos x]/[cos]^{2}

Step 3:Differentiate the indicated functions from Step 2. In this example, those functions are [sinx(x)] and [cos x].

f'(x)= cos^{2}(x) + sin^{2}(x) / cos^{2}x.

Step 4:**Use algebra** to simplify where possible. The solution is 1/cos^{2}(x), which is equivalent in trigonometry to sec^{2}(x).

## How to Differentiate 2^{x}/2^{x}x

In this example problem, you’ll need to know the algebraic rule that states:

a^{x}a^{x} = a^{x + x} = a^{2x} and a^{x}b^{x} = (ab)^{x}.

Step 1: **Name** the top term f(x) and the bottom term g(x).

- f(x) = 2
^{x} - g(x) = 2
^{x}– 3^{x}.

Step 2: **Place the functions** f(x) and g(x) from Step 1 into the **quotient rule**. The term d/dx here indicates a derivative.

f'(x) = (2^{x} – 3^{x}) d/dx[2^{x}] – (2^{x}) d/dx[2^{x} – 3^{x}]/(2^{x} –

3^{x})^{2}.

Step 3: Differentiate the indicated functions (d/dx)from Step 2. In this example, those functions are 2^{x} and [2^{x} – 3^{x}]

f'(x)= (2^{x} – 3^{x}) d/dx[2^{x} ln 2] – (2^{x})(2^{x}2^{x} ln 2 – 3^{x} ln 3).

Step 4:** Use algebra** to simplify where possible (remembering the rules from the intro).

f'(x) = 2^{2x} ln 2 – 6^{x} ln 2 – (2^{2x} ln 2 – 6^{x} ln 3) / (2^{x} – 3^{x})^{2}

By simplification, this becomes:

f'(x) = 6^{x}(ln 3 – ln 2) / (2^{x}-3^{x})^{2}

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