Derivative of e^-x^3

We now understand the derivative of e^-x^3. It is commonly represented as d/dx (e^-x^3) or (e^-x^3)', and its value is -3x²e^-x^3. The function e^-x^3 has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below:

Exponential Function: (e^u, where u = -x^3).

Chain Rule: Rule for differentiating e^-x^3 since it involves a composite function.

Power Rule: Used in conjunction with the chain rule to differentiate -x^3.

The derivative of e^-x^3 can be denoted as d/dx (e^-x^3) or (e^-x^3)'.

The formula we use to differentiate e^-x^3 is: d/dx (e^-x^3) = -3x²e^-x^3

The formula applies to all real x.

We can derive the derivative of e^-x^3 using proofs. To show this, we will use differentiation rules. There are several methods we use to prove this, such as:

1. Using Chain Rule
2. Using Product Rule

We will demonstrate that the differentiation of e^-x^3 results in -3x²e^-x^3 using these methods:

Using Chain Rule

To prove the differentiation of e^-x^3 using the chain rule, Consider f(x) = e^u where u = -x^3. Using the chain rule, d/dx [e^u] = e^u · du/dx.

Let's differentiate u = -x^3: du/dx = -3x².

Substituting back, d/dx (e^-x^3) = e^-x^3 · (-3x²) = -3x²e^-x^3.

Hence, proved.

Using Product Rule

The product rule isn't directly applicable to e^-x^3 as it is a single function, not a product of two functions, so we rely on the chain rule here.

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^3.

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^3, 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).

When x = 0, the derivative of e^-x^3 is -3(0)²e^0, which is 0. For large positive or negative x, the exponential term e^-x^3 rapidly approaches 0, making the derivative approach 0 as well.

• Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.

• Exponential Function: A function of the form e^u, where e is the base of natural logarithms.

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

• Power Rule: A basic rule in calculus used to find the derivative of a power function.

• Quotient Rule: A method of finding the derivative of a quotient of two functions.