Derivative of 6/x

We now understand the derivative of 6/x. It is commonly represented as d/dx (6/x) or (6/x)', and its value is -6/x². The function 6/x has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below: 

• Function Representation: (6/x is equivalent to 6·x⁻¹). 
   
• Power Rule: Rule for differentiating functions of the form axⁿ. 
   
• Negative Exponents: Understanding that x⁻¹ is equivalent to 1/x.

The derivative of 6/x can be denoted as d/dx (6/x) or (6/x)'.The formula we use to differentiate 6/x is: d/dx (6/x) = -6/x² (or) (6/x)' = -6/x²

The formula applies to all x where x ≠ 0.

We can derive the derivative of 6/x 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:

• By First Principle
   
• Using Power Rule

By First Principle

The derivative of 6/x can be proven using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of 6/x using the first principle, we will consider f(x) = 6/x. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = 6/x, we write f(x + h) = 6/(x + h). Substituting these into equation (1), f'(x) = limₕ→₀ [6/(x + h) - 6/x] / h = limₕ→₀ [(6x - 6(x + h)) / (x(x + h))] / h = limₕ→₀ [-6h / (x² + xh)] / h = limₕ→₀ [-6 / (x² + xh)] As h approaches 0, the expression becomes: f'(x) = -6/x² Hence, proved.

Using Power Rule

To prove the differentiation of 6/x using the power rule, We rewrite the function as 6·x⁻¹ and apply the power rule: d/dx [6·x⁻¹] = 6·(-1)x⁻² = -6/x² Thus, the derivative of 6/x is -6/x².

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a bit complex. To understand them better, consider a scenario where certain quantities change at different rates. Higher-order derivatives help analyze such functions effectively.

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 6/x, we generally use fⁿ(x) for the nth derivative of a function f(x) to understand the rate of change at different levels.

When x = 0, the derivative is undefined because 6/x has a vertical asymptote there.

When x = 1, the derivative of 6/x = -6/(1)², which is -6.

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

• Power Rule: A basic differentiation rule used to find the derivative of a function of the form axⁿ. 

• Undefined Points: Points where a function does not have a value, often due to division by zero.

• Rate of Change: A measure of how a quantity changes with respect to another variable. 

• Asymptote: A line that a curve approaches but never touches or intersects.