Derivative of Cube Root

We now understand the derivative of the cube root function. It is commonly represented as d/dx (∛x) or (∛x)', and its value is 1/(3∛(x²)). The function ∛x has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below:

Cube Root Function: The cube root of x is represented as ∛x.

Power Rule: Rule for differentiating functions of the form xⁿ.

Chain Rule: Used when differentiating functions that are composite.

The derivative of ∛x can be denoted as d/dx (∛x) or (∛x)'. The formula we use to differentiate ∛x is: d/dx (∛x) = 1/(3∛(x²)) The formula applies to all x except where x=0.

We can derive the derivative of ∛x using proofs. To show this, we will use the power rule along with the rules of differentiation. There are several methods we use to prove this, such as:

1. By First Principle
2. Using Chain Rule
3. Using Power Rule

We will now demonstrate that the differentiation of ∛x results in 1/(3∛(x²)) using the above-mentioned methods:

By First Principle

The derivative of ∛x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of ∛x using the first principle, we will consider f(x) = ∛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) = ∛x, we write f(x + h) = ∛(x + h). Substituting these into equation (1), f'(x) = limₕ→₀ [∛(x + h) - ∛x] / h

We simplify using the binomial expansion for roots: = limₕ→₀ [ (x + h)^(1/3) - x^(1/3) ] / h

Using the limit definition, we find: f'(x) = 1/(3x^(2/3))

Hence, proved.

Using Chain Rule

To prove the differentiation of ∛x using the chain rule, We express the cube root as a power: ∛x = x^(1/3). Using the chain rule, the derivative of x^(1/3) is: d/dx (x^(1/3)) = (1/3)x^(-2/3) Which is equivalent to: 1/(3∛(x²))

Using Power Rule

The power rule states if y = x^n, then dy/dx = nx^(n-1). For ∛x, rewrite it as x^(1/3).

Applying the power rule, d/dx (x^(1/3)) = (1/3)x^(-2/3) This simplifies to: 1/(3∛(x²))

Thus, it is 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 ∛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 ∛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 ∛x is not differentiable at x=0. When x is 1, the derivative of ∛x = 1/(3∛(1²)), which is 1/3.

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

• Cube Root Function: The cube root function is represented as ∛x, indicating the value which gives x when cubed.

• Power Rule: A rule used to differentiate functions of the form xⁿ.

• Chain Rule: A method for differentiating composite functions.

• Undefined Point: A point where the function is not differentiable or does not exist.