1. The problem is to find the derivative of a function, which means determining the rate at which the function's value changes with respect to its input variable. 2. The derivative of a function $f(x)$ is denoted as $f'(x)$ or $\frac{df}{dx}$ and is defined by the limit: $$ \frac{df}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} $$ This formula calculates the slope of the tangent line to the function at any point $x$. 3. Important rules for derivatives include: - The power rule: $\frac{d}{dx} x^n = n x^{n-1}$ - The sum rule: $\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)$ - The product rule: $\frac{d}{dx} [f(x) g(x)] = f'(x) g(x) + f(x) g'(x)$ - The quotient rule: $\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x) g(x) - f(x) g'(x)}{[g(x)]^2}$ - The 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 depending on the function's form. 5. For example, if $f(x) = x^3$, using the power rule: $$ f'(x) = 3 x^{3-1} = 3 x^2 $$ This means the derivative of $x^3$ is $3x^2$, which tells us how $x^3$ changes as $x$ changes. 6. In summary, the derivative measures how a function changes and is found using the limit definition and derivative rules.