Derivative of \(-e^{-x}\)

We now understand the derivative of \(-e^{-x}\).

It is commonly represented as \(\frac{d}{dx} (-e^{-x})\) or \((-e^{-x})\)', and its value is \(e^{-x}\).

The function \(-e^{-x}\) has a clearly defined derivative, indicating it is differentiable for all real x.

The key concepts are mentioned below:

Exponential Function: The exponential function is written as \(e^{x}\).

Chain Rule: A rule for differentiating composite functions like \(-e^{-x}\).

Negative Exponential: \(-e^{-x}\) involves a negative exponent, which flips the graph vertically.

The derivative of \(-e^{-x}\) can be denoted as \(\frac{d}{dx} (-e^{-x}\)) or (\(-e^{-x}\)).

The formula we use to differentiate \(-e^{-x}\) is: \(\frac{d}{dx} (-e^{-x}) = e^{-x}\)

This formula applies to all real x.

We can derive the derivative of \(-e^{-x}\) using proofs.

To show this, we will use the rules of differentiation along with the chain rule.

Let's demonstrate the differentiation of \(-e^{-x}\) resulting in \(e^{-x}\) using the chain rule:

Using Chain Rule

To prove the differentiation of \(-e^{-x}\) using the chain rule,

Consider \(f(x) = -e^{-x}\) We can rewrite it as \(f(x) = -u\) where \(u = e^{-x}\).

By the chain rule:

\(\frac{d}{dx} f(x) = \frac{d}{du}(-u) \cdot \frac{d}{dx}(e^{-x}\))  

\(\frac{d}{du}(-u) = -1\) and \(\frac{d}{dx}(e^{-x}) = -e^{-x}\)

Thus, \(\frac{d}{dx}(-e^{-x}) = -1 \cdot (-e^{-x}) = e^{-x}\)

Hence, proved.

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 \(-e^{-x}\).

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 \(-e^{-x}\), 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 derivative \(-e^{-x}\) is defined for all real numbers, which means there are no points where the derivative is undefined. When x = 0, the derivative of \(-e^{-x}\) is equal to \(e^{0} = 1\).

• 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 denoted as \(e^{x}\), where e is the base of the natural logarithm.

• Chain Rule: A rule for differentiating composite functions, crucial for functions like \(-e^{-x}\).

• Quotient Rule: A rule used to differentiate ratios of functions.

• Negative Exponential: A function involving a negative exponent, such as \(-e^{-x}\).