1. State the problem: Differentiate the function $f(x) = \frac{x^5}{x^2}$ using the quotient rule. 2. Formula: The quotient rule states that if $f(x) = \frac{g(x)}{h(x)}$, then $$f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}$$ 3. Important rule: When differentiating powers of $x$, use the power rule: $\frac{d}{dx} x^n = nx^{n-1}$. 4. Intermediate work: - Let $g(x) = x^5$, so $g'(x) = 5x^4$. - Let $h(x) = x^2$, so $h'(x) = 2x$. - Apply quotient rule: $$f'(x) = \frac{5x^4 \cdot x^2 - x^5 \cdot 2x}{(x^2)^2} = \frac{5x^{6} - 2x^{6}}{x^{4}} = \frac{3x^{6}}{x^{4}}$$ 5. Simplify: $$f'(x) = 3x^{6-4} = 3x^{2}$$ 6. Final answer: The derivative of $f(x) = \frac{x^5}{x^2}$ is $$f'(x) = 3x^{2}$$ This process can be applied to other quotient problems involving exponents. Here are 10 practice questions: 1. Differentiate $\frac{x^7}{x^3}$. 2. Differentiate $\frac{x^4}{x^5}$. 3. Differentiate $\frac{x^{10}}{x^2}$. 4. Differentiate $\frac{x^6}{x^6}$. 5. Differentiate $\frac{x^8}{x^4}$. 6. Differentiate $\frac{x^9}{x^3}$. 7. Differentiate $\frac{x^{12}}{x^7}$. 8. Differentiate $\frac{x^5}{x^1}$. 9. Differentiate $\frac{x^{11}}{x^5}$. 10. Differentiate $\frac{x^{15}}{x^{10}}$. Try applying the quotient rule and power rule to each to find their derivatives.