Derivative of Exp

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 for all real numbers.

The key concepts are mentioned below: Exponential Function: exp(x) = e^x.

Derivative Rule: Rule for differentiating exp(x) (the derivative of e^x is e^x).

Growth Rate: The rate at which exp(x) increases as x increases.

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) (or) (exp(x))' = exp(x)

The formula applies to all real numbers x.

We can derive the derivative of exp(x) using various methods. To show this, we will use the rules of differentiation.

There are several methods to prove this, such as: Using the Limit Definition Using Chain Rule Using Exponential Rules We will now demonstrate that the differentiation of exp(x) results in exp(x) using the above-mentioned methods:

Using the Limit Definition The derivative of exp(x) can be proved using the limit definition, which expresses the derivative as the limit of the difference quotient.

To find the derivative of exp(x) using the limit definition, we consider f(x) = exp(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) = exp(x), we write f(x + h) = exp(x + h).

Substituting these into equation (1), 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 the fact that limₕ→₀ [exp(h) - 1] / h = 1, f'(x) = exp(x) * 1 = exp(x)

Hence, proved. Using Chain Rule To prove the differentiation of exp(x) using the chain rule, We recognize that exp(x) is its own derivative: Let u = x, and f(u) = exp(u) Then d/dx [exp(x)] = d/du [exp(u)] * du/dx Since d/du [exp(u)] = exp(u) and du/dx = 1, d/dx [exp(x)] = exp(x)

Using Exponential Rules The function f(x) = exp(x) is unique in that its derivative is the same as the function itself.

We use the exponential rule: d/dx (e^x) = e^x This directly gives us: d/dx (exp(x)) = exp(x)

When a function is differentiated multiple times, the derivatives obtained are referred to as higher-order derivatives.

Higher-order derivatives of exp(x) are straightforward since they remain the same.

For the first derivative of a function, we write f′(x), which shows the function's change or slope at a certain point.

The second derivative is derived from the first derivative, which is denoted using f′′(x) and remains exp(x). Similarly, the third derivative, f′′′(x), is also exp(x), and this pattern continues.

For the nth derivative of exp(x), we generally use fⁿ(x), and it remains exp(x), indicating the change in the rate of change.

The exponential function exp(x) is defined and differentiable for all real numbers without any exceptions. For x = 0, the derivative of exp(x) = exp(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 mathematical function involving the constant e, typically written as exp(x) or e^x.

• Chain Rule: A rule for finding the derivative of a composition of functions.

• Product Rule: A rule that gives the derivative of the product of two functions.

• Quotient Rule: A rule that provides the derivative of the ratio of two functions.