1. The problem is to list the basic derivative formulas used in calculus. 2. The derivative of a function measures how the function's output changes as its input changes. 3. Here are the fundamental derivative formulas: - Constant Rule: The derivative of a constant $c$ is zero. $$\frac{d}{dx}[c] = 0$$ - Power Rule: For any real number $n$, the derivative of $x^n$ is: $$\frac{d}{dx}[x^n] = nx^{n-1}$$ - Constant Multiple Rule: The derivative of a constant times a function is the constant times the derivative of the function. $$\frac{d}{dx}[cf(x)] = c\frac{d}{dx}[f(x)]$$ - Sum Rule: The derivative of a sum is the sum of the derivatives. $$\frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$$ - Difference Rule: The derivative of a difference is the difference of the derivatives. $$\frac{d}{dx}[f(x) - g(x)] = \frac{d}{dx}[f(x)] - \frac{d}{dx}[g(x)]$$ - Product Rule: For two functions $f(x)$ and $g(x)$, $$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$$ - Quotient Rule: For two functions $f(x)$ and $g(x)$, $$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$$ - Chain Rule: For a composite function $f(g(x))$, $$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$$ These formulas are the foundation for finding derivatives of most functions encountered in calculus.