Exercise 2.5 — Differentiation Rules for Algebraic Functions

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

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Key Concepts

This exercise focuses on applying the following differentiation rules to algebraic (polynomial and rational) functions:

• Power Rule — differentiating $x^n$ and terms with negative exponents

• Product Rule — differentiating products of two functions

• Quotient Rule — differentiating rational functions

• Chain Rule — differentiating composite functions

• Evaluating derivatives at specific points

• Simplification strategies — expanding or splitting fractions before differentiating

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Important Formulas

Power Rule

$\frac{d}{dx}(x^n)=n{x}^{n-1}$

Also applies to negative exponents: $\frac{d}{dx}(x^{-n})=-n{x}^{-n-1}$

Product Rule

$\frac{d}{dx}[u(x)v(x)]=u^{\prime }(x)v(x)+u(x)v^{\prime }(x)$

Quotient Rule

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

Chain Rule

$\frac{d}{dx}[f(g(x))]=f^{\prime }(g(x))\cdot g^{\prime }(x)$

For the special case $y=\frac{k}{g(x)}=k[g(x){]}^{-1}$: $\frac{dy}{dx}=-k[g(x){]}^{-2}\cdot g^{\prime }(x)=-\frac{k{g}^{\prime }(x)}{[g(x){]}^2}$

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Key Strategies

Strategy 1: Simplify Before Differentiating

When the denominator is a monomial (e.g., $x^2$), split the fraction first: $y=\frac{x^4+2x^3-1}{x^2}=x^2+2x-x^{-2}$ Then apply the Power Rule term-by-term. This is often simpler than the Quotient Rule.

Strategy 2: Expand Products Before Differentiating

For products of polynomials, expanding first and using the Power Rule can be faster than the Product Rule: $y=(x+3)(x-3)=x^2-9⟹\frac{dy}{dx}=2x$

Strategy 3: Evaluating at a Point

To find $f^{\prime }(a)$: first find the general derivative $f^{\prime }(x)$, then substitute $x=a$.

Example: If $y=x^3-5x+2$, find $\frac{dy}{dx}$ at $x=2$. $\frac{dy}{dx}=3x^2-5⟹{\frac{dy}{dx}|}_{x=2}=3(4)-5=7$

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Summary

This exercise covers fundamental differentiation techniques for algebraic functions:

• Questions 1–10: Apply the Power, Product, and Quotient Rules to polynomials and rational functions.
• Questions 11–13: Use the Chain Rule for composite functions in denominators and squared terms.
• Questions 14–17: Combine differentiation with evaluation at specific points.

Key strategies: simplify expressions before differentiating when possible (e.g., Q10, Q14), and carefully track signs and exponents when applying the Quotient Rule. Watch for opportunities to expand products before differentiating as an alternative to the Product Rule.