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