Derivative of e to the -x

We now understand the derivative of e^(-x). It is commonly represented as d/dx (e^(-x)) or (e^(-x))', and its value is -e^(-x). The function e^(-x) has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Exponential Function: (e^(-x) is the exponential function with a negative exponent). Chain Rule: Rule for differentiating e^(-x) due to its composite nature. Negative Exponent Rule: Differentiation involving a function raised to a negative power.

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

We can derive the derivative of e^(-x) using proofs. To show this, we will use the chain rule along with the rules of differentiation. There are several methods we use to prove this, such as: Using Chain Rule We will now demonstrate that the differentiation of e^(-x) results in -e^(-x) using the chain rule: Using Chain Rule To prove the differentiation of e^(-x) using the chain rule, we use the formula: e^(-x) = e^(u), where u = -x By the chain rule: d/dx [e^(u)] = e^(u) * u' Let’s substitute u = -x, so u' = d/dx (-x) = -1 d/dx (e^(-x)) = e^(-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 e^(-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 e^(-x), 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 x is extremely large (approaching positive infinity), the derivative approaches zero since e^(-x) approaches zero. When x is 0, the derivative of e^(-x) = -e^(0), which is -1.

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) or e^(-x), where e is the base of the natural logarithm. Chain Rule: A rule in calculus used to differentiate composite functions. Product Rule: A rule in calculus used to differentiate products of two functions. Quotient Rule: A rule in calculus used to differentiate ratios of two functions.