Derivative of e^f(x)

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

The key concepts are mentioned below:

Exponential Function: (e^f(x) )where e is the base of the natural logarithm).

Chain Rule: Rule for differentiating e^f(x) (since it consists of an inner function f(x)).

Natural Logarithm: ln(x), which is the inverse of the exponential function.

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

The formula we use to differentiate e^f(x) is: d/dx (e^f(x)) = e^f(x) * f'(x) The formula applies to all x where f(x) is differentiable.

We can derive the derivative of e^f(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:

Using Chain Rule

To prove the differentiation of e^f(x) using the chain rule, We use the formula: e^f(x) = e^(u), where u = f(x) By the chain rule: d/dx [e^u] = e^u * du/dx Let’s substitute u = f(x) and du/dx = f'(x), d/dx (e^f(x)) = e^f(x) * f'(x)

Using First Principle

The derivative of e^f(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^f(x) using the first principle, we consider f(x) = e^f(x). Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [e^(f(x + h)) - e^f(x)] / h

Using the property of exponents, we rewrite it as: f'(x) = limₕ→₀ e^f(x) * [e^{(f(x + h)} - f(x)) - 1] / h Assuming f(x) is differentiable, we can use the limit definition of the derivative of f(x): f'(x) = e^f(x) * limₕ→₀ [f(x + h) - f(x)] / h f'(x) = e^f(x) * f'(x) Thus, the derivative of e^f(x) is e^f(x) * f'(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 e^f(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^f(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 f(x) is a constant function, the derivative of e^f(x) simplifies to 0 because f'(x) = 0. When f(x) is a linear function, the derivative of e^f(x) = e^f(x) * constant.

• 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 the form of e^x or e^f(x), where e is the base of the natural logarithm.

• Chain Rule: A rule used to differentiate composite functions by differentiating the outer function and multiplying by the derivative of the inner function.

• Product Rule: A rule used to differentiate the product of two functions.

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