Derivative of u

We now understand the derivative of u. It is commonly represented as d/dx (u) or (u)', and its derivative depends on the specific function u represents.

If u is a differentiable function, its derivative will be well-defined within its domain.

The key concepts to keep in mind include the rules of differentiation and how they apply to various forms of functions.

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

The specific formula for differentiating u depends on the form of the function u. For instance, if u = tan x, then d/dx (u) = sec²x.

The formula applies to all x where the function u is defined and differentiable.

We can derive the derivative of u using various proofs depending on the function u. To demonstrate this, we use differentiation rules and identities. Some methods include:

By First Principle - The derivative of a function like u can be shown using the First Principle, which expresses the derivative as the limit of the difference quotient.

Using Chain Rule-The chain rule helps in differentiating composite functions.

Using Product Rule-The product rule is used when u is the product of two or more functions. Here, we will demonstrate the differentiation of specific functions like tan x as an example:

By First Principle

For f(x) = tan x, the derivative can be expressed as the following limit: f'(x) = limₕ→₀ [f(x + h) - f(x)] / h Given f(x) = tan x, we write f(x + h) = tan(x + h).

Substituting these into the equation, we simplify and find f'(x) = sec²x.

Using Chain Rule

For tan x = sin x/cos x, applying the chain and quotient rule gives: d/dx(tan x) = sec²x.

Using Product Rule

For tan x = sin x · (cos x)⁻¹, we apply the product rule and find the derivative to be sec²x.

When a function is differentiated multiple times, the resulting derivatives are called higher-order derivatives. These can be complex to understand.

Consider a car, where the speed changes (first derivative) and the rate at which the speed changes (second derivative) also changes.

Higher-order derivatives help us understand more about functions like u. For example, if u = tan x, the first derivative is f′(x), and subsequent derivatives can be found similarly.

Special cases arise when the function u has undefined points or behaves irregularly. For instance, if u = tan x, the derivative is undefined at x = π/2 due to a vertical asymptote. At x = 0, the derivative is sec²(0), which equals 1.

• Derivative: Indicates how a function changes with a slight change in x.

• First Principle: A method to derive a function's derivative as a limit.

• Chain Rule: A rule for differentiating composite functions.

• Quotient Rule: A rule for differentiating a function that is a ratio of two functions.

• Product Rule: A rule used for differentiating the product of two functions.