Derivative of 1/e^x

We now understand the derivative of 1/e^x. It is commonly represented as d/dx (1/e^x) or (1/e^x)', and its value is -1/e^x. The function 1/e^x has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

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

Negative Exponent Rule: Used to simplify 1/e^x as e^-x.

Chain Rule: Rule for differentiating composite functions like e^-x.

The derivative of 1/e^x can be denoted as d/dx (1/e^x) or (1/e^x)'.

The formula we use to differentiate 1/e^x is: d/dx (1/e^x) = -1/e^x (or) (1/e^x)' = -1/e^x The formula applies to all x as e^x is always positive.

We can derive the derivative of 1/e^x using proofs. To show this, we will use the rules of differentiation and exponent rules.

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 1/e^x results in -1/e^x using the above-mentioned methods:

By First Principle The derivative of 1/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 1/e^x using the first principle, consider f(x) = 1/e^x. Its derivative can be expressed as the following limit.

f'(x) = limₕ→₀ [f(x + h) - f(x)] / h Given that f(x) = 1/e^x, we write f(x + h) = 1/e^(x + h).

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

Using Chain Rule To prove the differentiation of 1/e^x using the chain rule, We use the formula:

1/e^x = e^-x

The derivative of e^-x is found using the chain rule: d/dx (e^-x) = -e^-x

Therefore, d/dx (1/e^x) = -1/e^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 1/e^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 1/e^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).

The function 1/e^x has no undefined points as e^x is never zero. When x is 0, the derivative of 1/e^x = -1/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: A mathematical function of the form e^x, where e is the base of natural logarithms.

• Chain Rule: A fundamental rule in calculus used to differentiate composite functions.

• Negative Exponent: A mathematical operation that represents reciprocal powers, such as 1/e^x being e^-x.

• Rate of Change: A measure of how a quantity changes concerning another variable, often time.