Implicit Differentiation:
1. Find \( \frac{dy}{dx} \) by implicit differentiation.
(a) \(\displaystyle{\sqrt{x}-3\sqrt{y}=5} \) solution
(b) \(\displaystyle{ 2x^3-6y^3=8 } \)
solution
(c) \(\displaystyle{\frac{1}{x^3} + \frac{1}{y^2}=2 }\)
solution
(d) \(\displaystyle{x y - y^3 = 4x-1} \) solution
(e) \(\displaystyle{ x^2-xy+y^2= 5} \)
solution
(f) \(\displaystyle{e^{x+y} = 1+ x^2 y^2 } \)
solution
(g) \(\displaystyle{xy = \sqrt{x^2+y^2} } \)
solution
(h) \(\displaystyle{ y \cos x=x^2+y^2}\) solution
(i) \(\displaystyle{ y \cos(x^2)=x \cos(y^2)}\) solution
(j) \(\displaystyle{x^3y^2+ x \tan y = 2} \) solution
(k) \(\displaystyle{ 2-x=\sin(xy^2)}\) solution
(l) \(\displaystyle{\tan(x-y)=\frac{y}{1+x^2}}\) solution
2. Find the slope of the tangent line to the curve \( \tan(xy)
= y \) at the point \(\left(\frac{\pi}{4}, 1 \right)\). solution
3. Use implicit differentiation to find an equation of the tangent line to the curve
\(x^2 + xy + y^2=3 \) at the point \( (1,1) \). solution
Inverse Trigonometric Functions: Find the derivative of
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) \(\displaystyle{y=\sin^{-1}(5x^2+3) }\) (b) \(\displaystyle{y=5\tan^{-1} \sqrt{3x-1} }\) (c) \(\displaystyle{y=\tan^{-1} (e^x\sin x) }\) solution