Derivative of -e^x

We now understand the derivative of -e^x.

It is commonly represented as 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 across its domain.

The key concepts are mentioned below:

Exponential Function: (-e^x is an exponential function).

Derivative of e^x: The derivative of e^x is e^x.

Negative Constant Multiplier Rule: The derivative of a constant times a function is the constant times the derivative of the function.

The derivative of -e^x can be denoted as d/dx(-e^x) or (-e^x)'. The formula we use to differentiate -e^x is: d/dx(-e^x) = -e^x (or) (-e^x)' = -e^x This formula applies to all x in the domain of real numbers.

We can derive the derivative of -e^x using proofs.

To show this, we will use the properties of exponential functions along with the basic rules of differentiation. There are several methods we use to prove this, such as:

By First Principle Using Constant Multiplier Rule

We will now demonstrate that the differentiation of -e^x results in -e^x using the above-mentioned methods:

By First Principle The derivative of -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 -e^x using the first principle, we will consider f(x) = -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) = -e^x, we write f(x + h) = -e^{(x + h)}.

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

Using the limit properties, limₕ→₀ (e^h - 1)/h = 1, f'(x) = -e^x * 1 = -e^x.

Hence, proved.

Using Constant Multiplier Rule

To prove the differentiation of -e^x using the constant multiplier rule, We use the formula:

The derivative of a constant times a function is the constant times the derivative of the function.

Let u(x) = e^x, so -e^x = -1 * u(x).

By the constant multiplier rule: d/dx(-1 * u(x)) = -1 * d/dx(u(x)) Since d/dx(e^x) = e^x, we have: d/dx(-e^x) = -1 * e^x = -e^x. Thus, the derivative is -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 -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 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).

The exponential function -e^x is defined for all real x, hence it does not have any undefined points in its domain. When x is 0, the derivative of -e^x is -1, since -e⁰ = -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 a^x, where a is a constant. -e^x is a specific exponential function.

• First Derivative: The initial result of differentiating a function, showing the rate of change of the function.

• Constant Multiplier Rule: A rule stating that the derivative of a constant times a function is the constant times the derivative of the function.

• Quotient Rule: A rule used to differentiate functions that are the ratio of two differentiable functions.