Derivative of u/v

We now understand the derivative of u/v. It is commonly represented as d/dx (u/v) or (u/v)', and its value is given by the quotient rule. The function u/v has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below:

u and v Functions: u(x) and v(x) are functions of x.

Quotient Rule: Rule for differentiating u/v.

Derivative of a Function: The rate at which a function changes with respect to the independent variable.

The derivative of u/v can be denoted as d/dx (u/v) or (u/v)'.

The formula we use to differentiate u/v is: d/dx (u/v) = (v * u' - u * v') / v²

The formula applies to all x where v(x) ≠ 0.

We can derive the derivative of u/v 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 Product Rule
3. Using Chain Rule

We will now demonstrate that the differentiation of u/v results in (v * u' - u * v') / v² using the above-mentioned methods:

By First Principle

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

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

Substituting these into equation (1), f'(x) = limₕ→₀ [(u(x + h)/v(x + h)) - (u(x)/v(x))] / h = limₕ→₀ [(u(x + h) * v(x) - u(x) * v(x + h)) / (v(x) * v(x + h))] / h = limₕ→₀ [u(x + h) * v(x) - u(x) * v(x + h)] / [h * v(x) * v(x + h)] We apply the limit, f'(x) = (v * u' - u * v') / v²

Using Product Rule

To prove the differentiation of u/v using the product rule, We use the formula: u/v = u * (1/v) Consider u(x) and v(x) = v(x) Using the product rule: d/dx [u(x) * (1/v(x))] = u' * (1/v) + u * d/dx(1/v) Using the chain rule, d/dx(1/v) = -v'/v² Let’s substitute into the equation, d/dx(u/v) = u' * (1/v) - u * (v'/v²) Simplifying the expression, d/dx(u/v) = (v * u' - u * v') / v²

Using Chain Rule

We will now prove the derivative of u/v using the chain rule. The step-by-step process is demonstrated below: Consider u/v = u(x) * (v(x))⁻¹ Using the chain rule: d/dx [u(x) * (v(x))⁻¹] = u' * (v(x))⁻¹ - u * (v' * (v(x))⁻²) Simplifying, d/dx(u/v) = (v * u' - u * v') / v²

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 u/v.

For the first derivative of a function, we write f′(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 f′′(x). Similarly, the third derivative, f′′′(x), is the result of the second derivative, and this pattern continues.

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

When v(x) = 0, the derivative is undefined because u/v has a vertical asymptote there. When u(x) = 0, the derivative of u/v simplifies to zero.

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

• Quotient Rule: A rule for finding the derivative of a quotient of two functions.

• Chain Rule: A rule used to differentiate composite functions.

• First Derivative: It is the initial result of differentiating a function, which gives us the rate of change of a specific function.

• Asymptote: The line that a graph of a function approaches but never touches.