Find and simplify the derivative of the following functions.
f(x) = 3x^-9

Verified step by step guidance

1. Step 1: Identify the function for which you need to find the derivative. Here, the function is \( f(x) = 3x^{-9} \).
2. Step 2: Recall the power rule for differentiation, which states that if \( f(x) = ax^n \), then \( f'(x) = anx^{n-1} \).
3. Step 3: Apply the power rule to the function \( f(x) = 3x^{-9} \). Here, \( a = 3 \) and \( n = -9 \).
4. Step 4: Differentiate the function using the power rule: \( f'(x) = 3(-9)x^{-9-1} \).
5. Step 5: Simplify the expression obtained in Step 4: \( f'(x) = -27x^{-10} \).

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In calculus, the derivative is often denoted as f'(x) or df/dx, and it provides critical information about the function's behavior, such as its slope and points of tangency.

Power Rule

The Power Rule is a fundamental technique for finding the derivative of functions in the form f(x) = x^n, where n is any real number. According to this rule, the derivative is given by f'(x) = n*x^(n-1). This rule simplifies the differentiation process, especially for polynomial functions, and is essential for handling terms with negative or fractional exponents.

Simplification of Derivatives

After finding the derivative of a function, simplification is often necessary to express the result in its most concise form. This may involve combining like terms, reducing fractions, or applying algebraic identities. Simplifying the derivative not only makes it easier to interpret but also aids in further analysis, such as finding critical points or analyzing the function's behavior.