Differential Calculus: Practice Questions

Comprehensive practice covering all calculus topics - from limits to optimization

25 Questions

Full Chapter Coverage

How to Use This Practice Set

This comprehensive practice set covers all major topics in differential calculus. Questions are organized by topic using the tabs below. Each question includes helpful tips, visual diagrams where relevant, and detailed solutions. Work through each section systematically to build mastery!

Limits & First Principles

Master the foundation of calculus - limits and differentiation from first principles

Question 1

Limits

Evaluate $\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$.

Tip: When you get $\frac{0}{0}$, try factoring to eliminate the common term.

Solution:

\begin{align*} \lim_{x \to 3} \frac{x^2 - 9}{x - 3} &= \lim_{x \to 3} \frac{(x - 3)(x + 3)}{x - 3} \\ &= \lim_{x \to 3} (x + 3) \\ &= 3 + 3 \\ &= 6 \end{align*}

Answer: $6$

Question 2

First Principles

Use first principles to differentiate $f(x) = 3x^2 + 2$.

Remember: First principles uses $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

Solution:

\begin{align*} f'(x) &= \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \\ &= \lim_{h \to 0} \frac{[3(x + h)^2 + 2] - [3x^2 + 2]}{h} \\ &= \lim_{h \to 0} \frac{3(x^2 + 2xh + h^2) + 2 - 3x^2 - 2}{h} \\ &= \lim_{h \to 0} \frac{3x^2 + 6xh + 3h^2 - 3x^2}{h} \\ &= \lim_{h \to 0} \frac{6xh + 3h^2}{h} \\ &= \lim_{h \to 0} \frac{h(6x + 3h)}{h} \\ &= \lim_{h \to 0} (6x + 3h) \\ &= 6x \end{align*}

Answer: $f'(x) = 6x$

Ready for More?

Once you've mastered these practice questions, review the complete lesson or challenge yourself with the end-of-chapter quiz!

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