Derivative of 2x³

We now understand the derivative of 2x³. It is commonly represented as d/dx (2x³) or (2x³)', and its value is 6x². The function 2x³ has a well-defined derivative, which means it is differentiable across its entire domain.

The key concepts are mentioned below:

Polynomial Function: A function like 2x³ is a polynomial of degree 3.

Power Rule: The rule for differentiating terms like 2x³.

Coefficient: The number 2 in 2x³ is a constant multiplier.

The derivative of 2x³ can be denoted as d/dx (2x³) or (2x³)'. The formula we use to differentiate 2x³ is: d/dx (2x³) = 6x² (or) (2x³)' = 6x²

This formula applies to all x.

We can derive the derivative of 2x³ using several proofs. To show this, we will use basic differentiation rules.

There are various methods to prove this, such as:

• By First Principle
   
• Using Power Rule

We will now demonstrate that the differentiation of 2x³ results in 6x² using the mentioned methods:

By First Principle

The derivative of 2x³ can be found using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of 2x³ using the first principle, consider f(x) = 2x³. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = 2x³, we write f(x + h) = 2(x + h)³. Substituting these into equation (1), f'(x) = limₕ→₀ [2(x + h)³ - 2x³] / h = limₕ→₀ [2(x³ + 3x²h + 3xh² + h³) - 2x³] / h = limₕ→₀ [6x²h + 6xh² + 2h³] / h = limₕ→₀ [6x² + 6xh + 2h²] = 6x² (as h approaches 0) Hence, proved.

Using Power Rule

The power rule states that d/dx (xⁿ) = nxⁿ⁻¹. For 2x³, we have: d/dx (2x³) = 2 * d/dx (x³) Using the power rule: d/dx (x³) = 3x². Thus, d/dx (2x³) = 2 * 3x² = 6x².

When a function is differentiated multiple times, the resulting derivatives are referred to as higher-order derivatives. Higher-order derivatives can be complex. To understand them better, consider a car where speed changes (first derivative) and the acceleration changes (second derivative). Higher-order derivatives make it easier to understand functions like 2x³.

The first derivative of a function is written as f′(x), indicating how the function changes or its slope at a certain point. The second derivative is derived from the first derivative, denoted as f′′(x). Similarly, the third derivative, f′′′(x), results from the second derivative, and this pattern continues.

For the nth Derivative of 2x³, we generally use fⁿ(x) for the nth derivative of a function f(x), which tells us the change in the rate of change.

When x is 0, the derivative of 2x³ = 6(0)² = 0.

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

• Polynomial Function: A function composed of terms like axⁿ, such as 2x³.

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

• Coefficient: The constant term in front of a variable, such as 2 in 2x³.

• Second Derivative: The derivative of the derivative, providing information on the concavity of the original function.