Derivative of f/g

We now understand the derivative of f/g. It is commonly represented as d/dx (f/g) or (f/g)', and its value is given by the quotient rule: (f'g - fg')/g². The function f/g has a clearly defined derivative, indicating it is differentiable where g(x) ≠ 0. The key concepts are mentioned below:

Quotient Rule: A rule for differentiating a ratio of two functions (f/g).

Differentiation: The process of finding the derivative.

Function: A relation that uniquely associates members of one set with members of another set.

The derivative of f/g can be denoted as d/dx (f/g) or (f/g)'. The formula we use to differentiate f/g is: d/dx (f/g) = (f'g - fg')/g² The formula applies to all x where g(x) ≠ 0.

We can derive the derivative of f/g using proofs. To show this, we will use differentiation rules. There are several methods we use to prove this, such as:

1. By First Principle
2. Using Quotient Rule

We will now demonstrate that the differentiation of f/g results in (f'g - fg')/g² using the above-mentioned methods:

By First Principle

The derivative of f/g can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of f/g using the first principle, we will consider h(x) = f(x)/g(x). Its derivative can be expressed as the following limit. h'(x) = limₕ→₀ [h(x + h) - h(x)] / h … (1) Given that h(x) = f(x)/g(x), we write h(x + h) = f(x + h)/g(x + h).

Substituting these into equation (1), h'(x) = limₕ→₀ [f(x + h)/g(x + h) - f(x)/g(x)] / h = limₕ→₀ [f(x + h)g(x) - f(x)g(x + h)] / [h g(x)g(x + h)]

We now use the formula f(x + h)g(x) - f(x)g(x + h) = f'(x)g(x) - f(x)g'(x), h'(x) = limₕ→₀ [f'(x)g(x) - f(x)g'(x)] / [g²(x)]

As h approaches 0, we have, h'(x) = (f'g - fg')/g²

Hence, proved.

Using Quotient Rule

To prove the differentiation of f/g using the quotient rule, We use the formula: h(x) = f(x)/g(x)

By quotient rule: d/dx [f(x) / g(x)] = [f'(x)g(x) - f(x)g'(x)] / [g(x)]²… (1)

Let’s substitute f(x) and g(x) in equation (1), d/dx (f/g) = [(f'g - fg')]/g²

Thus, the derivative of f/g is (f'g - fg')/g².

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little tricky.

To understand them better, think of a car where the speed changes (first derivative) and the rate at which the speed changes (second derivative) also changes. Higher-order derivatives make it easier to understand functions like f/g.

For the first derivative of a function, we write h′(x), which indicates how the function changes or its slope at a certain point. The second derivative is derived from the first derivative, which is denoted using h′′(x) Similarly, the third derivative, h′′′(x) is the result of the second derivative and this pattern continues.

For the nth Derivative of f/g, we generally use hⁿ(x) for the nth derivative of a function h(x) which tells us the change in the rate of change. (continuing for higher-order derivatives).

When g(x) = 0, the derivative is undefined because h(x) becomes undefined. When the x is such that g(x) ≠ 0, the derivative of f/g is (f'g - fg')/g², ensuring differentiability.

• Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.

• Quotient Rule: A rule used to differentiate a ratio of two functions.

• Differentiation: The process of finding the derivative of a function.

• Function: A relation that uniquely associates members of one set with members of another set.

• Undefined: A term used when a function does not have a meaningful value at a certain point.