Derivative of x⁴

We now understand the derivative of x⁴. It is commonly represented as d/dx (x⁴) or (x⁴)', and its value is 4x³.

The function x⁴ has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Polynomial Function: (x⁴ is a polynomial function).

Power Rule: Rule for differentiating x⁴ (since it involves raising x to a power).

Coefficient: The constant that multiplies the variable x in the derivative.

The derivative of x⁴ can be denoted as d/dx (x⁴) or (x⁴)'. The formula we use to differentiate x⁴ is: d/dx (x⁴) = 4x³ (or) (x⁴)' = 4x³ The formula applies to all x in the domain of the real numbers.

We can derive the derivative of x⁴ using proofs.

To show this, we will use algebraic manipulation along with the rules of differentiation.

There are several methods we use to prove this, such as:

By First Principle

Using Power Rule

Using Product Rule

We will now demonstrate that the differentiation of x⁴ results in 4x³ using the above-mentioned methods:

By First Principle

The derivative of x⁴ can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of x⁴ using the first principle, we will consider f(x) = x⁴.

Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1)

Given that f(x) = x⁴, we write f(x + h) = (x + h)⁴.

Substituting these into equation (1), f'(x) = limₕ→₀ [(x + h)⁴ - x⁴] / h = limₕ→₀ [x⁴ + 4x³h + 6x²h² + 4xh³ + h⁴ - x⁴] / h = limₕ→₀ [4x³h + 6x²h² + 4xh³ + h⁴] / h = limₕ→₀ [4x³ + 6x²h + 4xh² + h³]

As h approaches 0, we have, f'(x) = 4x³.

Hence, proved.

Using Power Rule

To prove the differentiation of x⁴ using the power rule,

We use the formula: d/dx (xⁿ) = n*xⁿ⁻¹

For x⁴, n = 4, so: d/dx (x⁴) = 4*x³

This is a straightforward application of the power rule.

Using Product Rule

We will now prove the derivative of x⁴ using the product rule.

The step-by-step process is demonstrated below:

Here, we rewrite x⁴ as x*x*x*x.

Using the product rule formula:

d/dx [u.v] = u'.v + u.v'

Consider u = x and v = x³. u' = d/dx (x) = 1 v' = d/dx (x³) = 3x²

Using the product rule: d/dx (x⁴) = u'.v + u.v' = 1*x³ + x*3x² = x³ + 3x³ = 4x³

Thus: d/dx (x⁴) = 4x³.

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 x⁴.

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 x⁴, 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 x⁴ = 4x³, which is 0. When x is 1, the derivative of x⁴ = 4x³, which is 4. When x is -1, the derivative of x⁴ = 4x³, which is -4.

• 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 variables and coefficients involving only non-negative integer powers.

• Power Rule: A basic rule in calculus used for differentiating powers of x, expressed as d/dx(xⁿ) = n*xⁿ⁻¹.

• Coefficient: A numerical or constant factor in a term of an algebraic expression.

• Higher-Order Derivative: Derivatives of a function taken multiple times, providing information about the curvature and changes in the rate of change of the function.