Derivative of 3x³

We now understand the derivative of 3x³. It is commonly represented as d/dx (3x³) or (3x³)', and its value is 9x². The function 3x³ has a clearly defined derivative, indicating it is differentiable for all real numbers.

The key concepts are mentioned below:

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

Constant Multiple Rule: A rule for differentiating a constant times a function.

Polynomial Function: A function consisting of terms that are non-negative integer powers of x.

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

The formula we use to differentiate 3x³ is: d/dx (3x³) = 9x²

The formula applies to all real numbers x.

We can derive the derivative of 3x³ using proofs. To show this, we will use the basic rules 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 3x³ results in 9x² using the above-mentioned methods:

By First Principle

The derivative of 3x³ can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of 3x³ using the first principle, we will consider f(x) = 3x³. Its derivative can be expressed as the following limit: f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = 3x³, we write f(x + h) = 3(x + h)³. Substituting these into equation (1), f'(x) = limₕ→₀ [3(x + h)³ - 3x³] / h = limₕ→₀ [3(x³ + 3x²h + 3xh² + h³) - 3x³] / h = limₕ→₀ [9x²h + 9xh² + 3h³] / h = limₕ→₀ [9x² + 9xh + 3h²] = 9x² (as h approaches 0, the terms with h vanish) Hence, proved.

Using Power Rule

To prove the differentiation of 3x³ using the power rule, We use the formula: d/dx (xⁿ) = nxⁿ⁻¹ For 3x³, we consider the constant multiple rule: d/dx (3x³) = 3 * d/dx (x³) = 3 * 3x² = 9x² Thus, the derivative is 9x².

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 3x³.

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 3x³, 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 of 3x³ = 9x², which is 0. There are no undefined points in the domain of 3x³, as it is differentiable for all real numbers.

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

• Power Rule: A fundamental rule for differentiating functions of the form xⁿ, where n is a constant.

• Constant Multiple Rule: A rule that states the derivative of a constant times a function is the constant times the derivative of the function.

• Product Rule: A rule used to find the derivative of the product of two functions.

• Quotient Rule: A rule used to find the derivative of the division of two functions.