Derivative of xe^x

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

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

We can derive the derivative of xe^x using proofs. To show this, we will use differentiation rules. There are several methods we use to prove this, such as: Using Product Rule Using First Principle Using Product Rule To prove the differentiation of xe^x using the product rule, We use the formula: xe^x = x(e^x) The product rule states: d/dx [u·v] = u'·v + u·v' Let u = x and v = e^x u' = d/dx (x) = 1 v' = d/dx (e^x) = e^x Substituting into the product rule formula: d/dx (xe^x) = 1·e^x + x·e^x = e^x + xe^x = (x + 1)e^x Using First Principle The derivative of xe^x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of xe^x using the first principle, we will consider f(x) = xe^x. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h Given that f(x) = xe^x, we write f(x + h) = (x + h)e^(x + h). Substituting these into the equation, f'(x) = limₕ→₀ [(x + h)e^(x + h) - xe^x] / h = limₕ→₀ [xe^(x + h) + he^(x + h) - xe^x] / h = limₕ→₀ [xe^(x + h) - xe^x + he^(x + h)] / h = limₕ→₀ [e^x(x + h - x) / h + he^x] = limₕ→₀ [e^x + he^x/h] = e^x + e^x = (x + 1)e^x 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^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 xe^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 is 0, the derivative of xe^x = (0 + 1)e^0, which is 1. When x tends to negative infinity, the derivative approaches 0 as e^x becomes very small.

Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. Exponential Function: A function of the form e^x, where e is Euler's number, a mathematical constant approximately equal to 2.71828. Product Rule: A rule used to find the derivative of the product of two functions. First Derivative: It is the initial result of a function, which gives us the rate of change of a specific function. Chain Rule: A rule used to differentiate compositions of functions.