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 can be considered as a reciprocal function. Power Rule: A rule for differentiating functions of the form xⁿ. Negative Coefficient: The function -4/x has a negative coefficient that impacts its derivative.

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 (or) (-4/x)' = 4/x² The formula applies to all x where x ≠ 0.

We can derive the derivative of -4/x using proofs. To show this, we will use basic differentiation rules. Here are some methods we use to prove this: By Power Rule The function -4/x can be expressed as -4x⁻¹. Differentiate using the power rule: d/dx (xⁿ) = n*xⁿ⁻¹. For -4x⁻¹, n = -1: d/dx (-4x⁻¹) = (-1)(-4)x⁻² = 4x⁻² = 4/x². Hence, proved. By Quotient Rule Consider u = -4 and v = x. Apply the quotient rule: d/dx (u/v) = (v * du/dx - u * dv/dx) / v². Here, du/dx = 0 and dv/dx = 1: d/dx (-4/x) = (x*0 - (-4)*1) / x² = 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 how the acceleration of a car (second derivative) changes as the speed (first derivative) 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 = 0, the derivative is undefined because -4/x has a vertical asymptote there. When x is any non-zero real number, the derivative of -4/x = 4/x².

Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. Reciprocal Function: A function that is the inverse of another function, such as -4/x. Power Rule: A basic differentiation rule for functions of the form xⁿ. Undefined Point: A point where a function is not defined, such as x = 0 for -4/x. Chain Rule: A rule used to differentiate composite functions.