Subjects calculus

Derivative General Aef0C1

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Derivative General Aef0C1


1. The problem is to find the derivative of a function, but since the function is not specified, let's explain the general process of differentiation. 2. The derivative of a function $f(x)$, denoted as $f'(x)$ or $\frac{d}{dx}f(x)$, measures the rate at which $f(x)$ changes with respect to $x$. 3. The basic rules of differentiation include: - Power rule: $\frac{d}{dx} x^n = n x^{n-1}$ - Constant rule: $\frac{d}{dx} c = 0$ where $c$ is a constant - Sum rule: $\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)$ - Product rule: $\frac{d}{dx} [f(x) g(x)] = f'(x) g(x) + f(x) g'(x)$ - Quotient rule: $\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x) g(x) - f(x) g'(x)}{[g(x)]^2}$ - Chain rule: $\frac{d}{dx} f(g(x)) = f'(g(x)) g'(x)$ 4. To find the derivative of a specific function, apply these rules step-by-step to simplify and differentiate each term. Since no specific function was given, this is the general method to find the derivative. Final answer: The derivative depends on the function provided; use the above rules to compute it.