Exercise Questions

All questions in this exercise are listed below. Click on a question to view its solution.

Key Concepts

This exercise focuses on finding derivatives by first principles (the limit definition). Key topics include:

• 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

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Important Formulas

Below are the key formulas used in this exercise:

Derivative by First Principle (Definition): $f^{\prime }(x)={\lim }_{\Delta x\to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$

Slope of Tangent Line: $m_{\text{tangent}}=f^{\prime }(x)$

Power Rule: $\frac{d}{dx}\left[x^n\right]=n{x}^{n-1}$

Derivative of Cosine: $\frac{d}{dx}\left[\cos x\right]=-\sin x$

Chain Rule: $\frac{d}{dx}\left[f(g(x))\right]=f^{\prime }(g(x))\cdot g^{\prime }(x)$

Quotient Rule: $\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right]=\frac{u^{\prime }(x)v(x)-u(x)v^{\prime }(x)}{[v(x){]}^2}$

Velocity from Position Function: $v(t)=s^{\prime }(t)=\frac{d}{dt}[s(t)]$

Mean Value Theorem: $f^{\prime }(c)=\frac{f(b)-f(a)}{b-a},c\in (a,b)$

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Worked Example: Derivative by First Principle

Find $f^{\prime }(x)$ for $f(x)=x^2$ using the first principle.

Step 1: Write the difference quotient: $\frac{f(x+\Delta x)-f(x)}{\Delta x}=\frac{(x+\Delta x{)}^2-x^2}{\Delta x}$

Step 2: Expand the numerator: $=\frac{x^2+2x\Delta x+(\Delta x{)}^2-x^2}{\Delta x}=\frac{2x\Delta x+(\Delta x{)}^2}{\Delta x}$

Step 3: Cancel $\Delta x$: $=2x+\Delta x$

Step 4: Take the limit: $f^{\prime }(x)={\lim }_{\Delta x\to 0}(2x+\Delta x)=2x$

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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 using the first principle, 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 first principle, Power Rule, Chain Rule, and basic trigonometric derivatives is essential for success.