Derivative of 3y

We now understand the derivative of 3y. It is commonly represented as d/dy (3y) or (3y)', and its value is 3. The function 3y has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Constant Multiplication Rule: If a function is multiplied by a constant, its derivative is the constant multiplied by the derivative of the function.

Function of y: A function that depends on the variable y.

The derivative of 3y can be denoted as d/dy (3y) or (3y)'. The formula we use to differentiate 3y is: d/dy (3y) = 3 (or) (3y)' = 3 The formula applies to all y.

We can derive the derivative of 3y using proofs. To show this, we will use basic differentiation rules. There are several methods we use to prove this, such as:

1. Using the Constant Multiplication Rule
2. Using the Limit Definition of Derivatives
3. Using the Constant Multiplication Rule

To prove the differentiation of 3y using the constant multiplication rule, consider that if a constant 'c' multiplies a function f(y), its derivative is c times the derivative of f(y). Thus, if f(y) = y, then d/dy (f(y)) = 1.

Therefore, d/dy (3y) = 3 * d/dy (y) = 3 * 1 = 3.

Using the Limit Definition of Derivatives

To find the derivative of 3y using the limit definition, we consider f(y) = 3y. Its derivative can be expressed as the following limit: f'(y) = limₕ→₀ [f(y + h) - f(y)] / h

Given that f(y) = 3y, we write f(y + h) = 3(y + h).

Substituting these into the limit expression, f'(y) = limₕ→₀ [3(y + h) - 3y] / h = limₕ→₀ [3y + 3h - 3y] / h = limₕ→₀ 3h / h = limₕ→₀ 3 = 3.

Hence, proved.

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 3y.

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

For the nth Derivative of 3y, we generally use fⁿ(y) for the nth derivative of a function f(y), which tells us the change in the rate of change. (continuing for higher-order derivatives).

Since the function 3y is linear with respect to y, all higher-order derivatives beyond the first derivative are 0. When y = 0, the derivative of 3y = 3, which remains constant at any point on the line.

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

• Constant Multiplication Rule: A rule stating that the derivative of a constant times a function is the constant times the derivative of the function.

• Linear Function: A function of the form ax + b, where a and b are constants.

• Higher-Order Derivative: Derivatives of a function taken multiple times, such as the second or third derivative.

• Sum Rule: A rule that states the derivative of a sum of functions is the sum of their derivatives.