Derivative of x^1/3

We now understand the derivative of x^(1/3). It is commonly represented as d/dx (x^(1/3)) or (x^(1/3))', and its value is (1/3)x^(-2/3). The function x^(1/3) has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Power Rule: Rule for differentiating x^n where n is any real number. Negative Exponents: Understanding how exponents can be negative in derivatives.

The derivative of x^(1/3) can be denoted as d/dx (x^(1/3)) or (x^(1/3))'. The formula we use to differentiate x^(1/3) is: d/dx (x^(1/3)) = (1/3)x^(-2/3) The formula applies to all x where x is not 0.

We can derive the derivative of x^(1/3) 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^(1/3) results in (1/3)x^(-2/3) using the above-mentioned methods: By First Principle The derivative of x^(1/3) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of x^(1/3) using the first principle, we will consider f(x) = x^(1/3). 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^(1/3), we write f(x + h) = (x + h)^(1/3). Substituting these into equation (1), f'(x) = limₕ→₀ [(x + h)^(1/3) - x^(1/3)] / h Using binomial expansion or numerical methods to simplify, f'(x) = (1/3)x^(-2/3) Using Power Rule To prove the differentiation of x^(1/3) using the power rule, We use the formula: d/dx (x^n) = nx^(n-1) For x^(1/3), n = 1/3, so we apply the power rule: d/dx (x^(1/3)) = (1/3)x^(-2/3)

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^(1/3). 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^(1/3), 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 x^(1/3) is not differentiable at x = 0. When x is negative, the derivative is still valid since we can take odd roots of negative numbers, but care must be taken with the real domain.

Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. Power Rule: A basic rule used to find the derivative of a power of x. Negative Exponents: Exponents that indicate the reciprocal of the base raised to a positive exponent. First Derivative: It is the initial result of a function, which gives us the rate of change of a specific function. Undefined Point: A point where the function or its derivative does not exist.