ƒ¹(x) [xⁿ]
The notation ƒ¹(x) [xⁿ] represents the derivative of a function ƒ(x) with respect to x raised to the nth power
The notation ƒ¹(x) [xⁿ] represents the derivative of a function ƒ(x) with respect to x raised to the nth power. In other words, it denotes the nth derivative of ƒ(x) with respect to x.
To understand this notation better, let’s break it down step by step:
1. ƒ(x): This represents a function of x. It can be any mathematical expression involving x, such as f(x) = x^2, f(x) = sin(x), or even more complex functions like f(x) = e^(2x).
2. The superscript ¹: This denotes the first derivative. If we represent the first derivative of ƒ(x) as ƒ'(x), then ƒ¹(x) is equivalent to ƒ'(x). This derivative measures the rate at which the function ƒ(x) changes with respect to x.
3. The brackets [xⁿ]: This indicates that we raise the derivative (in this case, the first derivative) to the power of n. The exponent represents the number of times we want to differentiate the function. For example, if n = 2, then [x²] means taking the second derivative, or if n = 3, then [x³] represents the third derivative, and so on.
To summarize, ƒ¹(x) [xⁿ] refers to the nth derivative of ƒ(x) with respect to x. It represents differentiating the function ƒ(x) n times and then raising that derivative to the nth power.
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