Derivative of x^-1/2

We now understand the derivative of x^-1/2. It is commonly represented as d/dx (x^-1/2) or (x^-1/2)', and its value is -1/2 * x^-3/2. The function x^-1/2 has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Power Function: (x^n where n = -1/2). Power Rule: Rule for differentiating x^n. Negative Exponent: A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent.

The derivative of x^-1/2 can be denoted as d/dx (x^-1/2) or (x^-1/2)'. The formula we use to differentiate x^-1/2 is: d/dx (x^-1/2) = -1/2 * x^-3/2 The formula applies to all x where x > 0.

We can derive the derivative of x^-1/2 using proofs. To show this, we will use the power rule of differentiation. There are several methods we use to prove this, such as: By First Principle Using Power Rule We will now demonstrate that the differentiation of x^-1/2 results in -1/2 * x^-3/2 using these methods: By First Principle The derivative of x^-1/2 can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of x^-1/2 using the first principle, we will consider f(x) = x^-1/2. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = x^-1/2, we write f(x + h) = (x + h)^-1/2. Substituting these into equation (1), f'(x) = limₕ→₀ [(x + h)^-1/2 - x^-1/2] / h Using algebraic manipulation to combine the fractions, = limₕ→₀ [(x^-1/2(1 - (x + h)^1/2/x^1/2))] / h As h approaches zero, we get the derivative, f'(x) = -1/2 * x^-3/2. Hence, proved. Using Power Rule To prove the differentiation of x^-1/2 using the power rule, We use the formula: d/dx (x^n) = n * x^(n-1). Substituting n = -1/2 in the power rule, d/dx (x^-1/2) = (-1/2) * x^(-1/2 - 1) = -1/2 * x^-3/2. Thus, the derivative of x^-1/2 is -1/2 * x^-3/2.

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 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 x^-1/2. 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 x^-1/2, we generally use f^(n)(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 x^-1/2 has a vertical asymptote there. When x is 1, the derivative of x^-1/2 = -1/2 * 1^-3/2, which is -1/2.

Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. Power Rule: A differentiation rule used for functions of the form x^n. Negative Exponent: An exponent that indicates the reciprocal of the base raised to the absolute value of the exponent. Reciprocal: A value that, when multiplied by the original number, results in 1. Asymptote: A line that a curve approaches without crossing.