Derivative of x⁴
2026-02-28 23:51 Diff
https://brightchamps.com/en-us/math/calculus/derivative-of-x%5E4

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.