Derivative of e

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 across its entire domain. The key concepts are mentioned below: Exponential Function: e^x is the exponential function where the base is the mathematical constant e. Constant Function Rule: Differentiation of e^x uses the rule that the derivative of a constant multiplied by a function is the constant times the derivative of the function. Chain Rule: Used for differentiating composite functions involving e^x.

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 The formula applies to all x within the real number set.

We can derive the derivative of e^x using proofs. To show this, we will use the properties of exponential functions along with the rules of differentiation. There are several methods we use to prove this, such as: By First Principle Using Chain Rule Using Limit Definitions We will now demonstrate that the differentiation of e^x results in e^x using the above-mentioned methods: By First Principle The derivative of e^x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of e^x using the first principle, consider f(x) = e^x. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = e^x, we write f(x + h) = e^(x + h). Substituting these into equation (1), f'(x) = limₕ→₀ [e^(x + h) - e^x] / h = limₕ→₀ [e^x * e^h - e^x] / h = limₕ→₀ [e^x (e^h - 1)] / h We now use the fact that limₕ→₀ (e^h - 1)/h = 1. f'(x) = e^x * 1 = e^x Hence, proved. Using Chain Rule To prove the differentiation of e^x using the chain rule, Consider y = e^x. Then, if we have a composite function, y = e^(g(x)), we differentiate using: dy/dx = e^(g(x)) * g'(x) For the simpler case y = e^x, g(x) = x, so g'(x) = 1. Thus, dy/dx = e^x * 1 = e^x. Using Limit Definitions We can also use the limit definition of the exponential function. The limit definition of e is given by: e = limₙ→∞ (1 + 1/n)ⁿ Using this, we can derive the derivative: d/dx (e^x) = e^x * limₕ→₀ [(1 + 1/n)^n - 1]/h = e^x Thus the derivative of e^x is e^x.

When a function is differentiated multiple times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives of e^x are straightforward to understand because the derivative of e^x remains e^x through successive differentiations. 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 denote it as fⁿ(x) = e^x for all n, indicating that the exponential function maintains its form through differentiation.

When x approaches negative or positive infinity, the behavior of e^x is exponential growth or decay. 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: An exponential function is a mathematical function of the form e^x, where e is the constant approximately equal to 2.71828. Chain Rule: A rule for differentiating compositions of functions, used when a function is composed of an outer function and an inner function. First Principle: A method of finding a derivative using the limit of the difference quotient. Higher-Order Derivative: Successive derivatives of a function, representing rates of change of rates of change.