Derivative of 2x²

We now understand the derivative of 2x². It is commonly represented as d/dx (2x²) or (2x²)', and its value is 4x. The function 2x² has a clearly defined derivative, indicating it is differentiable for all real numbers.

The key concepts are mentioned below:

Polynomial Function: A function consisting of terms with variables raised to whole number powers.

Power Rule: A basic rule for differentiating functions of the form ax^n.

Constant Multiple Rule: If a function is multiplied by a constant, the derivative is the constant multiplied by the derivative of the function.

The derivative of 2x² can be denoted as d/dx (2x²) or (2x²)'. The formula we use to differentiate 2x² is: d/dx (2x²) = 4x (or) (2x²)' = 4x. The formula applies to all x in the domain of real numbers.

We can derive the derivative of 2x² using proofs. To show this, we will use basic differentiation rules. There are several methods we use to prove this, such as:

1. By First Principle
2. Using Power Rule

We will now demonstrate that the differentiation of 2x² results in 4x using the above-mentioned methods:

By First Principle

The derivative of 2x² can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of 2x² using the first principle, we will consider f(x) = 2x².

Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = 2x², we write f(x + h) = 2(x + h)².

Substituting these into equation (1), f'(x) = limₕ→₀ [2(x + h)² - 2x²] / h = limₕ→₀ [2(x² + 2xh + h²) - 2x²] / h = limₕ→₀ [2x² + 4xh + 2h² - 2x²] / h = limₕ→₀ [4xh + 2h²] / h = limₕ→₀ 4x + 2h = 4x (as h approaches 0).

Hence, proved.

Using Power Rule

To prove the differentiation of 2x² using the power rule, We use the formula: d/dx (x^n) = n * x^(n-1)

For 2x², we consider it as 2 * x². By the constant multiple rule, we take the constant out and differentiate x². d/dx (2x²) = 2 * d/dx (x²) = 2 * (2x) = 4x.

Hence, 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 2x².

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 2x², 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 2x² = 4x, which is 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 consisting of terms with variables raised to whole number powers.

• Power Rule: A basic rule for differentiating functions of the form ax^n, giving n * ax^(n-1).

• Constant Multiple Rule: If a function is multiplied by a constant, the derivative is the constant multiplied by the derivative of the function.

• Higher-Order Derivatives: Derivatives obtained by differentiating a function multiple times, providing insights into the function's behavior.