Derivative of exp(x)

We now understand the derivative of exp(x). It is commonly represented as d/dx (exp(x)) or (exp(x))', and its value is exp(x). The function exp(x) has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Exponential Function: (exp(x)). Basic Differentiation Rule: Rule for differentiating exp(x) directly.

The derivative of exp(x) can be denoted as d/dx (exp(x)) or (exp(x))'. The formula we use to differentiate exp(x) is: d/dx (exp(x)) = exp(x) The formula applies to all x.

We can derive the derivative of exp(x) using proofs. To show this, we will use the definition of the derivative along with the properties of exponential functions. There are several methods we use to prove this, such as: By First Principle Using Properties of Exponents We will now demonstrate that the differentiation of exp(x) results in exp(x) using the above-mentioned methods: By First Principle The derivative of exp(x) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of exp(x) using the first principle, we will consider f(x) = exp(x). Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h Given that f(x) = exp(x), we write f(x + h) = exp(x + h). Substituting these into the equation, f'(x) = limₕ→₀ [exp(x + h) - exp(x)] / h = limₕ→₀ [exp(x) * exp(h) - exp(x)] / h = exp(x) * limₕ→₀ [exp(h) - 1] / h Using limit properties, limₕ→₀ [exp(h) - 1] / h = 1. f'(x) = exp(x) * 1 = exp(x) Hence, proved. Using Properties of Exponents To prove the differentiation of exp(x) using properties of exponents, We use the property: exp(x + h) = exp(x) * exp(h) The derivative, according to the first principle, is: f'(x) = limₕ→₀ [exp(x) * exp(h) - exp(x)] / h = exp(x) * limₕ→₀ [exp(h) - 1] / h Using the known limit, limₕ→₀ [exp(h) - 1] / h = 1. f'(x) = exp(x) * 1 = exp(x) Thus, the derivative of exp(x) is exp(x).

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 exp(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 exp(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).

The derivative of exp(x) is always exp(x) for any real number x, meaning there are no special cases of undefined points like vertical asymptotes.

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 in the form of exp(x), which represents continuous growth or decay. Chain Rule: A method used in calculus to differentiate compositions of functions. Product Rule: A differentiation rule used to find the derivative of the product of two functions. Quotient Rule: A method in calculus for finding the derivative of a division of two functions.