Derivative of 1/(x+2)

We now understand the derivative of 1/(x+2). It is commonly represented as d/dx (1/(x+2)) or (1/(x+2))', and its value is -1/(x+2)². The function 1/(x+2) has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Rational Function: A function of the form 1/(x+2).

Quotient Rule: Rule for differentiating 1/(x+2) (as it can be seen as 1 divided by a function of x).

Power Rule: Used to differentiate functions of the form xⁿ.

The derivative of 1/(x+2) can be denoted as d/dx (1/(x+2)) or (1/(x+2))'. The formula we use to differentiate 1/(x+2) is: d/dx (1/(x+2)) = -1/(x+2)²

The formula applies to all x except where x = -2, as this would make the denominator zero.

We can derive the derivative of 1/(x+2) using proofs. To show this, we will use algebraic manipulation along with the rules of differentiation.

There are several methods we use to prove this, such as:

1. By First Principle
2. Using Chain Rule
3. Using Power Rule

We will now demonstrate that the differentiation of 1/(x+2) results in -1/(x+2)² using the above-mentioned methods:

By First Principle

The derivative of 1/(x+2) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of 1/(x+2) using the first principle, we will consider f(x) = 1/(x+2). Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1)

Given that f(x) = 1/(x+2), we write f(x + h) = 1/(x+h+2).

Substituting these into equation (1), f'(x) = limₕ→₀ [1/(x+h+2) - 1/(x+2)] / h = limₕ→₀ [(x+2) - (x+h+2)] / [(x+2)(x+h+2)h] = limₕ→₀ [-h] / [(x+2)(x+h+2)h] = limₕ→₀ -1 / [(x+2)(x+h+2)]

As h approaches 0, we simplify the expression: f'(x) = -1 / (x+2)²Hence, proved.

Using Chain Rule

To prove the differentiation of 1/(x+2) using the chain rule, We use the formula: 1/(x+2) = (x+2)⁻¹

Let u = x+2

Thus, y = u⁻¹

Using the chain rule formula: d/dx [uⁿ] = nuⁿ⁻¹ du/dx dy/dx = -1 * (x+2)⁻² * d/dx (x+2) dy/dx = -1/(x+2)² Hence, proved.

Using Power Rule

We will now prove the derivative of 1/(x+2) using the power rule. The step-by-step process is demonstrated below:

Rewrite the function: y = (x+2)⁻¹

Using the power rule to differentiate: d/dx (y) = -1 * (x+2)⁻² * d/dx (x+2) d/dx (y) = -1/(x+2)²

Thus, the derivative of 1/(x+2) is -1/(x+2)².

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 acceleration (second derivative) as the rate of change of speed (first derivative). Higher-order derivatives make it easier to understand functions like 1/(x+2).

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 1/(x+2), 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 x is -2, the derivative is undefined because 1/(x+2) has a vertical asymptote there. When x is 0, the derivative of 1/(x+2) = -1/(0+2)², which is -1/4.

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

• Rational Function: A function that can be expressed as the ratio of two polynomials.

• Chain Rule: A rule for differentiating compositions of functions.

• Quotient Rule: A rule used to differentiate functions expressed as the quotient of two functions.

• Power Rule: A rule used to differentiate functions of the form xⁿ.