# **Exercise Questions** All questions in this exercise are listed below. Click on a question to view its solution. --- --- # **Key Concepts** This exercise focuses on the following concepts: - Slope of the tangent line to a curve at a point - Derivative as the instantaneous rate of change - Power Rule for differentiation - Differentiation of trigonometric functions (cosine) - Chain Rule and Quotient Rule for composite and rational functions - Mean Value Theorem for functions on closed intervals - Applications of derivatives to physics: position, velocity, and acceleration - Projectile motion and free-fall problems --- # **Important Formulas** Below are the key formulas used in this exercise: **Slope of Tangent Line:** $$m_{\text{tangent}} = f'(x)$$ **Power Rule:** $$\frac{d}{dx}\left[x^n\right] = nx^{n-1}$$ **Derivative of Cosine:** $$\frac{d}{dx}\left[\cos x\right] = -\sin x$$ **Chain Rule:** $$\frac{d}{dx}\left[f(g(x))\right] = f'(g(x)) \cdot g'(x)$$ **Quotient Rule:** $$\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$$ **Velocity from Position Function:** $$v(t) = s'(t) = \frac{d}{dt}[s(t)]$$ **Mean Value Theorem:** $$f'(c) = \frac{f(b) - f(a)}{b - a}$$ --- # **Summary** This exercise develops the fundamental concept of the derivative as the slope of the tangent line, progressing from basic linear and polynomial functions to rational, trigonometric, and composite functions. Problems 1-6 establish the geometric interpretation of the derivative, while Problems 7-8 introduce the Mean Value Theorem for analyzing average and instantaneous rates of change over intervals. Problems 9-12 apply these techniques to physical contexts, connecting mathematical derivatives to the concepts of velocity and acceleration in motion problems. Mastery of the Power Rule, Chain Rule, and basic trigonometric derivatives is essential for success.