Derivative of 4/x

We now understand the derivative of 4/x.

It is commonly represented as d/dx (4/x) or (4/x)', and its value is -4/x².

The function 4/x has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below: Reciprocal Function: (4/x is a constant divided by x).

Power Rule: Rule for differentiating x raised to a power. Quotient Rule: Used for differentiating expressions like 4/x.

The derivative of 4/x can be denoted as d/dx (4/x) or (4/x)'.

The formula we use to differentiate 4/x is: d/dx (4/x) = -4/x² The formula applies to all x ≠ 0.

We can derive the derivative of 4/x using proofs. To show this, we will use basic differentiation rules.

There are several methods we use to prove this, such as: Using the Power Rule Using the Quotient Rule We will now demonstrate that the differentiation of 4/x results in -4/x² using the above-mentioned methods: Using the Power Rule The derivative of 4/x can be expressed by rewriting it as 4x⁻¹.

To find the derivative, consider f(x) = 4x⁻¹. Using the power rule, which states that d/dx (xⁿ) = n·xⁿ⁻¹, f'(x) = 4(-1)x⁻² f'(x) = -4/x²

Hence, proved. Using the Quotient Rule To prove the differentiation of 4/x using the quotient rule, We use the formula: d/dx (u/v) = (v·u' - u·v') / v² Let u = 4 and v = x, u' = d/dx (4) = 0 v' = d/dx (x) = 1

Substituting these into the quotient rule, d/dx (4/x) = (x·0 - 4·1) / x² d/dx (4/x) = -4/x² 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 4/x. 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 4/x, 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 0, the derivative is undefined because the function 4/x is undefined at x = 0.

As x approaches infinity, the derivative of 4/x approaches 0, indicating the slope becomes flatter.

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

• Reciprocal Function: A function of the form k/x, where k is a constant.

• Power Rule: A basic differentiation rule used to find the derivative of x raised to a power.

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

• Undefined Points: Points at which a function does not exist or is not continuous.