Derivative of x+y

We now understand the derivative of x+y. It is commonly represented as d/dx (x+y) or (x+y)'. The derivative of x is 1, and the derivative of y is 1, assuming both x and y are differentiable with respect to their respective variables.

The key concepts are mentioned below:

Simple Addition: The function x+y is the sum of two variables.

Derivative of a Constant: The derivative of a constant is 0.

Linearity of Differentiation: Differentiation distributes over addition.

The derivative of x+y can be denoted as d/dx (x+y) or (x+y)'.

The formula we use to differentiate x+y is: d/dx (x+y) = 1 + 0 = 1 (or) (x+y)' = 1

Similarly, d/dy (x+y) = 0 + 1 = 1

This formula holds for all x and y where they are differentiable.

We can derive the derivative of x+y using basic differentiation rules. To show this, we will use the rules of differentiation, including:

1. By Linearity of Differentiation
2. Using Basic Derivative Rules

By Linearity of Differentiation

The derivative of x+y can be proved using the linearity of differentiation, which states that the derivative of a sum is the sum of the derivatives.

To find the derivative of x+y, we consider f(x, y) = x+y. Its derivative with respect to x can be expressed as: f'(x) = d/dx (x+y) = d/dx (x) + d/dx (y) = 1 + 0 = 1

Similarly, the derivative with respect to y is: f'(y) = d/dy (x+y) = d/dy (x) + d/dy (y) = 0 + 1 = 1

Hence, proved.

Using Basic Derivative Rules

To prove the differentiation of x+y using basic rules, Consider the functions f(x) = x and g(y) = y.

The derivatives are straightforward: d/dx (x) = 1 and d/dy (y) = 1 Thus, d/dx (x+y) = 1 and d/dy (x+y) = 1

Therefore, the derivative of x+y is 1 with respect to both x and y.

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be less complex for linear functions like x+y.

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, f′′(x), would be 0 for linear functions like x+y, as they don't have curvature. Similarly, the third derivative, f′′′(x), is also 0, and this pattern continues.

For the nth Derivative of x+y, we generally use fⁿ(x), where n ≥ 2, resulting in 0, indicating no change in curvature.

When x or y is a constant, the derivative of that constant is 0. The derivative of x or y with respect to itself is always 1.

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

• Linearity: The principle that the derivative of a sum is the sum of the derivatives.

• Constant: A value that does not change and whose derivative is 0.

• Quotient Rule: A rule used to differentiate expressions that are the quotient of two functions.

• Chain Rule: A rule used to differentiate compositions of functions.