1. The problem is to find the derivative of a function, which measures how the function changes as its input changes. 2. The derivative of a function $f(x)$ is denoted as $f'(x)$ or $\frac{d}{dx}f(x)$. 3. The basic rules for derivatives include: - Power rule: $\frac{d}{dx}x^n = nx^{n-1}$ - 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, apply these rules step-by-step depending on the function given. 5. For example, if $f(x) = 3x^2 + 5x - 4$, then: - Derivative of $3x^2$ is $3 \times 2x^{2-1} = 6x$ - Derivative of $5x$ is $5$ - Derivative of $-4$ is $0$ - So, $f'(x) = 6x + 5$ 6. This derivative tells us the slope of the tangent line to the curve at any point $x$.