Implicit Differentiation

1. Find \( \frac{dy}{dx} \) by implicit differentiation.

(a) \( \sqrt{x} - 3\sqrt{y} = 5 \)  solution

(b) \( 2x^3 - 6y^3 = 8 \)  solution

(c) \( \frac{1}{x^3} + \frac{1}{y^2} = 2 \)  solution

(d) \( xy - y^3 = 4x - 1 \)  solution

(e) \( x^2 - xy + y^2 = 5 \)  solution

(f) \( e^{x+y} = 1 + x^2 y^2 \)  solution

(g) \( xy = \sqrt{x^2 + y^2} \)  solution

(h) \( y\cos x = x^2 + y^2 \)  solution

(i) \( y\cos(x^2) = x\cos(y^2) \)  solution

(j) \( x^3 y^2 + x\tan y = 2 \)  solution

(k) \( 2 - x = \sin(xy^2) \)  solution

(l) \( \tan(x - y) = \frac{y}{1 + x^2} \)  solution

2. Tangent Line Problems

Find the slope of the tangent line to \( \tan(xy) = y \) at \( \left(\frac{\pi}{4}, 1\right) \).  solution

Use implicit differentiation to find the tangent line to \( x^2 + xy + y^2 = 3 \) at \( (1,1) \).  solution

Inverse Trigonometric Functions

1. (a) \( y = \sin^{-1}(e^x) \)   (b) \( y = \tan^{-1}(x^2 + 5x) \)  solution

2. \( f(x) = \sin^{-1}(x\tan x) \)  solution

3. (a) \( y = \sin^{-1}(5x^2 + 3) \)   (b) \( y = 5\tan^{-1}\sqrt{3x - 1} \)   (c) \( y = \tan^{-1}(e^x \sin x) \)  solution