Derivative of x^a

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

The key concepts are mentioned below:

Power Function: (x^a).

Power Rule: Rule for differentiating x^a.

Constant Function: A function that always returns the same value.

The derivative of x^a can be denoted as d/dx (x^a) or (x^a)'. The formula we use to differentiate x^a is: d/dx (x^a) = a*x^(a-1)

The formula applies to all x where x is not equal to 0 when a is negative.

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

• By First Principle Using the Chain Rule
   
• Using the Product Rule

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

By First Principle

The derivative of x^a can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of x^a using the first principle, we will consider f(x) = x^a. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h Given that f(x) = x^a, we write f(x + h) = (x + h)^a. Substituting these into the equation, f'(x) = limₕ→₀ [(x + h)^a - x^a] / h Using the binomial theorem to expand (x + h)^a, f'(x) = limₕ→₀ [x^a + a*x^(a-1)*h + higher order terms - x^a] / h = limₕ→₀ [a*x^(a-1) + higher order terms] = a*x^(a-1) Hence, proved. Using the Chain Rule To prove the differentiation of x^a using the chain rule, We consider y = x^a as a composite function of u = x and y = u^a.

By the chain rule:

dy/dx = dy/du * du/dx Given that du/dx = 1, we have: d/dx (x^a) = a*u^(a-1) * 1 = a*x^(a-1). Using the Product Rule We will now prove the derivative of x^a using the product rule. The step-by-step process is demonstrated below. Consider x^a as a product of x and x^(a-1). Using the product rule formula: d/dx [u*v] = u'v + uv' Let u = x and v = x^(a-1) u' = 1 and v' = (a-1)*x^(a-2) Thus, d/dx (x^a) = 1*x^(a-1) + x*(a-1)*x^(a-2) = x^(a-1) + (a-1)*x^(a-1) = a*x^(a-1).

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^a.

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^a, 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 a = 0, the derivative is 0 because x⁰ is a constant function. When x = 0 and a < 0, the derivative is undefined because x^a has a vertical asymptote there.

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

• Power Function: A function of the form x^a, where a is a constant.

• Power Rule: A basic rule in calculus used to differentiate functions of the form x^a.

• Constant Function: A function that always returns the same value, regardless of the input.

• Chain Rule: A rule for differentiating compositions of functions.