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Apuntes

**1–4.** **(a)** Find \( y' \) by implicit differentiation. **(b)** Solve the equation explicitly for \( y \) and differentiate to get \( y' \) in terms of \( x \). **(c)** Check that your solutions to parts (a) and (b) are consistent by substituting the expression for \( y \) into your solution for part (a). --- **5–22. Find \( \frac{dy}{dx} \) by implicit differentiation.** **5.** \( x^2 - 4xy + y^2 = 4 \) **6.** \( 2x^2 + xy - y^2 = 2 \) **7.** \( x^4 + x^2 y + y^3 = 5 \) **8.** \( x^3 - xy^2 + y^3 = 1 \) **9.** \( \frac{x^2}{x + y} = y^2 + 1 \) **10.** \( xe^x = x - y \) **11.** \( \sin x + \cos y = 2x - 3y \) **12.** \( e^x \sin y = x + y \) **13.** \( \sin(x + y) = \cos x + \cos y \) **14.** \( \tan(x - y) = 2xy^3 + 1 \) **15.** \( y \cos x = x^2 + y^2 \) **16.** \( \sin(xy) = \cos(x + y) \) **17.** \( 2xe^x + y e^{x} = 3 \) **18.** \( \sin x \cos y = x^2 - 5y \) **19.** \( \sqrt{x + y} = x^4 + y^4 \) **21.** \( \frac{e^{xy}}{x} = - y \) **22.** \( \cos^2 (x + y) = xe^y \) --- **23.** If \( f(x) + x^2 [f(x)]^3 = 10 \) and \( f(1) = 2 \), find \( f'(1) \). **24.** If \( g(x) + x \sin g(x) = x^2 \), find \( g'(0) \).