Derivative of xe^2x

We now understand the derivative of xe^2x. It is commonly represented as d/dx (xe^2x) or (xe^2x)', and its value is (2x + 1)e^2x. The function xe^2x has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Exponential Function: e^x is a base function used in the differentiation of xe^2x. Product Rule: Rule for differentiating xe^2x (since it consists of the product of x and e^2x). Constant Function: The number 2 is a constant in the function e^2x.

The derivative of xe^2x can be denoted as d/dx (xe^2x) or (xe^2x)'. The formula we use to differentiate xe^2x is: d/dx (xe^2x) = (2x + 1)e^2x The formula applies to all x.

We can derive the derivative of xe^2x using proofs. To show this, we will use differentiation rules. There are several methods we use to prove this, such as: Using the Product Rule Using the Chain Rule We will now demonstrate that the differentiation of xe^2x results in (2x + 1)e^2x using the above-mentioned methods: Using the Product Rule To prove the differentiation of xe^2x using the product rule, We use the formula: xe^2x = x·e^2x Consider f(x) = x and g(x) = e^2x 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) = e^2x in the product rule, d/dx (xe^2x) = (1)·e^2x + x·(2e^2x) d/dx (xe^2x) = e^2x + 2xe^2x d/dx (xe^2x) = (2x + 1)e^2x Hence, proved. Using the Chain Rule To prove the differentiation of xe^2x using the chain rule, Let y = xe^2x, then y' = d/dx [x·e^2x] Apply the product rule: y' = e^2x + x·d/dx[e^2x] y' = e^2x + x·(2e^2x) y' = (2x + 1)e^2x Hence, proved.

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 xe^2x. 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 xe^2x, 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 xe^2x is (2x + 1)e^2x, which simplifies to e^0 = 1.

Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. Exponential Function: A mathematical function involving the constant e, often seen in growth and decay problems. Product Rule: A fundamental rule in calculus used to differentiate products of two functions. Chain Rule: A rule used in calculus to differentiate composite functions. Higher-Order Derivative: A derivative that is obtained by differentiating a function multiple times.