Derivative of e^-5x

We now understand the derivative of e^-5x. It is commonly represented as d/dx (e^-5x) or (e^-5x)', and its value is -5e^-5x.
 

The function e^-5x has a clearly defined derivative, indicating it is differentiable everywhere. The key concepts are mentioned below:

Exponential Function: e^-5x is an exponential function with a base e and a constant exponent.

Chain Rule: A rule for differentiating compositions of functions, which is often used when differentiating exponential functions.

Constant Multiplier Rule: A rule used to differentiate functions multiplied by a constant.

The derivative of e^-5x can be denoted as d/dx (e^-5x) or (e^-5x)'. The formula we use to differentiate e^-5x is: d/dx (e^-5x) = -5e^-5x The formula applies to all x.

We can derive the derivative of e^-5x using several methods.
 

To show this, we will use the chain rule and the properties of exponential functions.

Here are the methods we use to prove this: Using Chain Rule To prove the differentiation of e^-5x using the chain rule, Consider u(x) = -5x and the outer function v(u) = e^u.

The derivative of v with respect to u is v'(u) = e^u, and the derivative of u with respect to x is u'(x) = -5. Using the chain rule: (v(u(x)))' = v'(u(x)) * u'(x), d/dx (e^-5x) = e^-5x * (-5) = -5e^-5x.

Hence, proved.

Using Constant Multiplier Rule We will now prove the derivative of e^-5x using the constant multiplier rule.

Given that d/dx (e^x) = e^x, For y = e^-5x, let u = -5x, then dy/du = e^u and du/dx = -5.

By the chain rule, dy/dx = dy/du * du/dx = e^u * (-5) = -5e^-5x. Thus, the derivative of e^-5x is -5e^-5x.

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^-5x. 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^-5x, the pattern can be observed as fⁿ(x) = (-5)ⁿe^-5x, indicating the change in the rate of change.

When x approaches infinity, e^-5x approaches zero, meaning the derivative -5e^-5x also approaches zero. When x is 0, the derivative of e^-5x = -5e^0, which is -5.

• 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 in which an independent variable appears in the exponent.

• Chain Rule: A rule for differentiating compositions of functions.

• Constant Multiplier Rule: A rule for differentiating functions multiplied by a constant.

• Quotient Rule: A rule for differentiating a quotient of two functions.