Derivative of x|x|

We now understand the derivative of x|x|. It is represented as d/dx (x|x|) and its value is |x| + x * (d/dx |x|). The function x|x| has a clearly defined derivative, indicating it is differentiable except at x = 0. The key concepts are mentioned below:

Absolute Value Function: |x|.

Piecewise Definition: |x| is defined as x for x ≥ 0 and -x for x < 0.

Derivative of Absolute Value: d/dx |x| = x/|x| for x ≠ 0.

The derivative of x|x| can be denoted as d/dx (x|x|).

The formula we use to differentiate x|x| is: d/dx (x|x|) = |x| + x * (d/dx |x|) = 2x for x > 0 and 0 for x < 0.

The formula applies to all x except x = 0, where |x| changes from negative to positive.

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

1. By First Principle
2. Using Chain Rule
3. Using Piecewise Differentiation

We will now demonstrate that the differentiation of x|x| results in 2x for x > 0 and 0 for x < 0 using the above-mentioned methods:

By First Principle

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

To find the derivative of x|x| using the first principle, we will consider f(x) = x|x|. Its derivative can be expressed as the following limit. f'(x) = lim(h→0) [f(x + h) - f(x)] / h

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

Substituting these into the equation, f'(x) = lim(h→0) [(x + h)|x + h| - x|x|] / h For x > 0, |x| = x, hence |x + h| = x + h

when h is small and positive. f'(x) = lim(h→0) [(x + h)(x + h) - x²] / h = lim(h→0) [x² + 2xh + h² - x²] / h = lim(h→0) [2xh + h²] / h = lim(h→0) [2x + h] = 2x For x < 0, |x| = -x,

hence |x + h| = -(x + h) when h is small. f'(x) = lim(h→0) [-(x + h)(x + h) - (-x)x] / h = lim(h→0) [-x² - 2xh - h² + x²] / h = lim(h→0) [-2xh - h²] / h = lim(h→0) [-2x - h] = 0 Hence, proved.

Using Chain Rule

To prove the differentiation of x|x| using the chain rule, We use the formula: x|x| = x * (x for x ≥ 0 and -x for x < 0) Consider f(x) = x and g(x) = |x|

By product rule: d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x) Let’s substitute f(x) = x and g(x) = |x| d/dx (x|x|) = 1 · |x| + x · (d/dx |x|) For x > 0, |x| = x and d/dx |x| = 1,

hence d/dx (x|x|) = x + x · 1 = 2x For x < 0, |x| = -x and d/dx |x| = -1, hence d/dx (x|x|) = -x + x · (-1) = 0

Using Piecewise Differentiation

We will now prove the derivative of x|x| using piecewise differentiation: Here, we use the formula, x|x| = { x², for x ≥ 0 -x², for x < 0 }

Differentiate each piece: For x ≥ 0: d/dx (x²) = 2x For x < 0: d/dx (-x²) = 0 Thus, d/dx (x|x|) = { 2x, for x > 0 0, for x < 0 }

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|x|.

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|x|, 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 = 0, the derivative is undefined because |x| is non-differentiable at 0. When x > 0, the derivative of x|x| = 2x. When x < 0, the derivative of x|x| = 0.

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

• Absolute Value: A function that gives the non-negative value of a number or expression, represented by |x|.

• Piecewise Function: A function defined by different expressions based on different intervals of the domain.

• First Principle: A fundamental approach to finding the derivative using limits.

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