Derivative of x³/3

We now understand the derivative of x³/3. It is commonly represented as d/dx (x³/3) or (x³/3)', and its value is x². The function x³/3 has a clearly defined derivative, indicating it is differentiable across its domain.

The key concepts are mentioned below:

Cubic Function: (x³/3 is a cubic polynomial).

Power Rule: Rule for differentiating x³/3.

Simplification: Simplifying the expression before differentiating can make the process straightforward.

The derivative of x³/3 can be denoted as d/dx (x³/3) or (x³/3)'.

The formula we use to differentiate x³/3 is: d/dx (x³/3) = x² The formula applies to all real numbers x.

We can derive the derivative of x³/3 using proofs. To show this, we will use basic calculus principles along with differentiation rules.

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 x³/3 results in x² using the above-mentioned methods:

By First Principle

The derivative of x³/3 can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of x³/3 using the first principle, we will consider f(x) = x³/3. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x+h) - f(x)] / h … (1) Given that f(x) = x³/3, we write f(x+h) = (x+h)³/3. Substituting these into equation (1), f'(x) = limₕ→₀ [(x+h)³/3 - x³/3] / h = limₕ→₀ [((x³ + 3x²h + 3xh² + h³)/3) - x³/3] / h = limₕ→₀ [3x²h + 3xh² + h³]/(3h) = limₕ→₀ [x² + xh + h²/3] = x² + limₕ→₀ [xh + h²/3] = x² + 0 Thus, f'(x) = x². Hence, proved.

Using Power Rule

To prove the differentiation of x³/3 using the power rule, Consider f(x) = x³/3 We know that d/dx (xⁿ) = n*xⁿ⁻¹ Therefore, d/dx (x³) = 3x² Now, consider x³/3 = (1/3)x³ Applying the constant multiple rule, d/dx [(1/3)x³] = (1/3) * d/dx (x³) = (1/3) * 3x² = x² Thus, d/dx (x³/3) = x².

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 x³/3.

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 x³/3, 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 = 0, the derivative is 0 because x² = 0. For any positive or negative value of x, the derivative x² will always be positive, indicating that the function is always increasing for x > 0 and decreasing for x < 0.

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

• Cubic Function: A polynomial function in which the highest degree term is cubed, such as x³.

• Power Rule: A basic rule in calculus used to find the derivative of a power function.

• Constant Multiple Rule: A rule that allows the differentiation of a constant multiple of a function.

• First Derivative: The initial result of a function, which gives us the rate of change of a specific function.