Derivative of e^ax

We now understand the derivative of e^ax. It is commonly represented as d/dx (e^ax) or (e^ax)', and its value is ae^ax. The function e^ax has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Exponential Function: (e^ax is an exponential function where e is the base of the natural logarithm).

Chain Rule: Rule for differentiating e^ax (since it involves a composite function).

Constant Multiple Rule: When differentiating constant multiples of functions.

The derivative of e^ax can be denoted as d/dx (e^ax) or (e^ax)'.

The formula we use to differentiate e^ax is: d/dx (e^ax) = ae^ax (or) (e^ax)' = ae^ax

The formula applies to all x.

We can derive the derivative of e^ax using proofs. To show this, we will use the exponential properties along with the rules of differentiation.

There are several methods we use to prove this, such as:

By First Principle Using Chain Rule We will now demonstrate that the differentiation of e^ax results in ae^ax using the above-mentioned methods:

By First Principle The derivative of e^ax can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of e^ax using the first principle, we will consider f(x) = e^ax. 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^ax, we write f(x + h) = e^(a(x + h)).

Substituting these into equation (1), f'(x) = limₕ→₀ [e^(a(x + h)) - e^ax] / h = limₕ→₀ [e^ax (e^(ah) - 1)] / h = e^ax limₕ→₀ [e^(ah) - 1] / h

We now make a substitution: let u = ah, then as h approaches 0, u approaches 0. f'(x) = e^ax limᵤ→₀ [e^u - 1] / (u/a) = ae^ax limᵤ→₀ [e^u - 1] / u

Using limit formulas, limᵤ→₀ [e^u - 1] / u = 1. f'(x) = ae^ax. Hence, proved.

Using Chain Rule To prove the differentiation of e^ax using the chain rule, We use the formula: Let u = ax Then e^ax = e^u

By chain rule: d/dx [e^u] = e^u · du/dx = e^ax · a Therefore, d/dx (e^ax) = ae^ax.

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^ax.

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^ax, 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 = 0, the derivative of e^ax = ae^(a*0), which is a. The derivative of e^0 is 0 because e^0 equals 1, and the derivative of a constant is 0.

• Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.

• Exponential Function: A function in which an independent variable appears in the exponent, e.g., e^ax.

• Chain Rule: A formula for computing the derivative of the composition of two or more functions. Constant

• Multiple Rule: When differentiating a constant multiplied by a function, the constant factor is kept. First

• Derivative: It is the initial result of a function, which gives us the rate of change of a specific function.