91Ó°ÊÓ

The power rule is a basic but powerful rule for differentiating functions of the form \(x^n\). This rule states that if you have a function \(y = x^n\), its derivative \(y'\) is computed as: \[ \frac{d}{dx}[x^n] = nx^{n-1}\. \] It’s straightforward: just bring down the exponent and subtract one from the original exponent.
To see this rule in action, let's use the function \(x^{-3}\). Applying the power rule, we get: \[ \frac{d}{dx}[x^{-3}] = -3x^{-4}.\ \] The rule easily adapts for any value of \(n\), no matter if \(n\) is positive, negative, or even a fraction.
This simplicity makes the power rule a go-to tool for differentiating polynomial functions.

Negative exponents are a way to express the reciprocal of a number or variable raised to a power. For example, \(x^{-3}\) is equivalent to \frac{1}{x^3}\. This conversion is essential for simplifying and differentiating functions using rules like the power rule.
In the given exercise, the original function, \frac{1}{3x^3}\, is rewritten using a negative exponent as \frac{1}{3}x^{-3}\. This allows us to apply differentiation techniques more straightforwardly.
Rewriting functions with negative exponents makes handling derivatives easier, especially when dealing with polynomial functions. Remember, converting back to positive exponents at the end of your derivation helps present your final answer more clearly.

Polynomial functions are expressions involving sums of powers of \(x\) with constant coefficients. Typical examples include functions like \3x^2 + 2x - 5\ or \x^4 + 4x^3 + x - 7\.
To differentiate polynomial functions, we often rely on the power rule combined with other derivative rules (like the sum rule). Here's a step-by-step breakdown:
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