1. The problem is to understand the definition of the derivative and its applications such as instantaneous velocity and tangent lines. 2. The derivative of a function $f$ at a point $x$ is defined as the limit: $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ This represents the instantaneous rate of change of $f$ at $x$. 3. For velocity, if $f(t)$ represents position at time $t$, then instantaneous velocity is: $$\lim_{h \to 0} \frac{f(t+h) - f(t)}{h} = f'(t)$$ 4. The equation of the tangent line to $f(x)$ at $x=a$ is: $$y = f(a) + f'(a)(x - a)$$ This is a linear approximation of $f$ near $a$. 5. The slope-intercept form of a line is: $$y = mx + b$$ where $m$ is the slope and $b$ is the y-intercept. 6. Important derivative rules: - Power Rule: For functions $f$ and $g$, the derivative of their product is: $$(tfg)' = f'g + fg'$$ - Quotient Rule: For $f/g$, the derivative is: $$\frac{d}{dx} \left( \frac{f}{g} \right) = \frac{f'g - fg'}{g^2}$$ These rules help compute derivatives of complex functions. 7. The spiral pattern mentioned likely relates to curves and their tangents in 3D or 2D, illustrating how derivatives describe slopes and rates of change along curves. Final answer: The derivative $f'(x)$ is the limit definition given, used to find instantaneous velocity and tangent lines, with formulas and rules as above.