Derivative of e^-8x

We now understand the derivative of e^-8x. It is commonly represented as d/dx (e^-8x) or (e^-8x)', and its value is -8e^-8x. The function e^-8x has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Exponential Function: (e^x is an exponential function). Chain Rule: A rule for differentiating compositions of functions (used here for e^-8x). Constant Multiplication Rule: The derivative of a constant times a function is the constant times the derivative of the function.

The derivative of e^-8x can be denoted as d/dx (e^-8x) or (e^-8x)'. The formula we use to differentiate e^-8x is: d/dx (e^-8x) = -8e^-8x The formula applies to all x.

We can derive the derivative of e^-8x using proofs. To show this, we will use the rules of differentiation. There are several methods we use to prove this, such as: Using Chain Rule Using Constant Multiplication Rule We will now demonstrate that the differentiation of e^-8x results in -8e^-8x using the above-mentioned methods: Using Chain Rule To prove the differentiation of e^-8x using the chain rule, We use the formula: Let u = -8x, then the function becomes e^u. The derivative of e^u with respect to u is e^u. The derivative of u = -8x with respect to x is -8. By chain rule: d/dx (e^u) = e^u * (du/dx) d/dx (e^-8x) = e^-8x * (-8) d/dx (e^-8x) = -8e^-8x Hence, proved. Using Constant Multiplication Rule We will now prove the derivative of e^-8x using the constant multiplication rule. Given that, d/dx (e^-8x) = d/dx (e^(-8x)) = -8 * d/dx (e^x) = -8 * e^x Substituting back, d/dx (e^-8x) = -8e^-8x Thus, the derivative of e^-8x is -8e^-8x.

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^-8x. 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^-8x, 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).

When x approaches infinity, the derivative approaches 0 because the exponential function decays. At x = 0, the derivative of e^-8x = -8e^0, which is -8.

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 involving an exponent, typically written as e^x for the natural exponential function. Chain Rule: A rule in calculus for differentiating compositions of functions. Constant Multiplication 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 method for finding the derivative of a quotient of two functions.