Derivative of Power of x

We now understand the derivative of x^n. It is commonly represented as d/dx (x^n) or (x^n)', and its value is n*x^(n-1). The function x^n has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

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

Polynomial Functions: Functions like x^n, where n is a non-negative integer.

The derivative of x^n can be denoted as d/dx (x^n) or (x^n)'. The formula we use to differentiate x^n is: d/dx (x^n) = n*x^(n-1) This formula applies to all x and for any real number n.

We can derive the derivative of x^n using proofs. To show this, we will use the properties of limits along with the rules of differentiation.

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

1. By First Principle
2. Using the Binomial Theorem

We will now demonstrate that the differentiation of x^n results in n*x^(n-1) using the above-mentioned methods:

By First Principle

The derivative of x^n can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of x^n

using the first principle, we will consider f(x) = x^n. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1)

Given that f(x) = x^n, we write f(x + h) = (x + h)^n.

Substituting these into equation (1), f'(x) = limₕ→₀ [(x + h)^n - x^n] / h

Using the Binomial Theorem, expand (x + h)^n: = limₕ→₀ [x^n + n*x^(n-1)*h + ... + h^n - x^n] / h = limₕ→₀ [n*x^(n-1)*h + ... + h^n] / h = limₕ→₀ [n*x^(n-1) + ... + h^(n-1)]

Using limit properties, all terms containing h vanish as h approaches 0, leaving: f'(x) = n*x^(n-1) Hence, proved.

Using Binomial Theorem

To prove the differentiation of x^n using the binomial theorem, We use the expansion: (x + h)^n = x^n + n*x^(n-1)*h + ... + h^n

By focusing on the terms that involve h, when we divide by h and take the limit as h approaches 0, we find that only the first-order term, n*x^(n-1), remains, yielding the derivative.

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 x^n.

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 x^n, we generally use f^(n)(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 n = 0, the derivative is undefined because x^0 = 1, which is a constant with a derivative of 0. When n = 1, the derivative of x^1 = 1*x^(1-1), which is 1.

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

• Power Rule: A rule for differentiating functions of the form x^n, resulting in n*x^(n-1).

• Polynomial Functions: Functions that consist of terms like x^n, where n is a non-negative integer.

• First Derivative: The initial result of a function’s differentiation, giving us the rate of change of a specific function.

• Binomial Theorem: A theorem that provides a formula for expanding powers of sums, used in differentiating powers of x.