# Let's see some quotient rule examples

First of all, you have to know that **the quotient rule** must be utilized when the derivative of the quotient of two functions is to be taken.

In words, the **quotient rule calculus** may seem a little bit complicated: the derivative of the quotient of 2 or more functions equals the denominator multiplied by the derivative of the numerator minus the numerator multiplied by the derivative of the denominator, and the whole ensemble divided by the square of the denominator. Obviously, the denominator should be different from zero.

Basically, the rule teaches us how to find the **quotient rule derivative**. We have two differentiable functions put as one function in a quotient form. The rule shows us the transformations the **quotient rule derivatives** suffer in order to become one single function. The following **quotient rule examples** will make you understand the fact much better.

(1) Find the derivative of (4x – 2)/(x^{2} + 1):

## Study of quotient rule formula

Now that you have seen the concrete example of how the rule works, it is the time to announce the **quotient rule formula.** It helps to differentiate the function of the form f(x) = g(x)/h(x) in the following manner.

f'(x) = [g'(x)h(x) – g(x)h'(x)] / [h(x)]^{2 }provided that the function h(x) is not equal to zero.

In order to make you understand how to use the **quotient rule for exponents**, we decided to take an example:

The **quotient rule for radicals** can be applied in this manner to solve the expressions. Try to solve the **quotient rule practice problems **to have a mastery over the subject.

## Lets see the quotient rule proof

The **quotient rule proof** is a little bit more difficult, but the starting point in the demonstration is the definition of the derivative itself. **Partial derivative quotient rule** is based on the followings: Suppose z = u/v, where u = x + y and v = x – y. As we can see, z depends partially on both x and y. This is why the term becomes to partial derivative and we can use the partial derivative quotient rule. The rule says that if we calculate the partial derivative of z with respect to x, we take y as a constant (or the reverse case).

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