6.3 Implicit Differentiation Review

Select all of the following that are implicitly defined functions.

• $4xy^{2\ }+3xy-4=12x$

• $y=2x^{3\ }-3x+1$

• $\sqrt{x}+2xy=y^3$

• $y=\sqrt{x+3}-11$

• $2xy^2-2x^2y=100$

Find  $\frac{dy}{dx}$  for the function  $5x^3=-5y^3+3$  

•  $\frac{-15x^2}{-5y^3+3}$  

•  $-\frac{x^2}{y^2}$  

•  $-\frac{y^2}{x^2}$  

•  $\frac{5x^3}{-5y^3+3}$  

Find  $\frac{dy}{dx}$  for the function  $-5y^3-4x^2y+3=3x^2$  

•  $\frac{-5y^3-4x^2y+3}{3x^2}$  

•  $\frac{6x+8xy}{-15y^2-4x^2}$  

•  $\frac{6+8y}{3x}$  

•  $\frac{-15y^2-4x^2}{6x+8xy}$  

Find  $\frac{d^2y}{dx^2}$  in terms of x and y for  $2y^2+1=5x$  

•  $-\frac{25}{16y^3}$  

•  $\frac{8y^3-10y^2+4y-5}{25x^2}$  

•  $\frac{16y}{25}$  

•  $-\frac{1}{x^2}$  

What is the slope of the tangent line to  $x=-2y^2+4$  at the point  $\left(2,\ 1\right)$ ?

(Type your answer with NO spaces!)

Write the equation of the tangent line to  $4x-4y=4x^3y$  at the point  $\left(0,\ 0\right)$  

Find the value of  $\frac{dy}{dx}$  for the function  $-2x^2+3=2x^3+3y^3$  at  $x=-1$  

•  $-\frac{9}{2}$  

•  $2$  

•  $1$  

•  $-\frac{2}{9}$  

State the slope of the normal line to $\left(x-1\right)^2+\left(y-1\right)^2=13$  
  at the point  $\left(3,\ 4\right)$  

(Type your answer with NO spaces!)

Write the equation of the normal line to the curve  $y^4-x^4-y^2+x^2=0$  at the point  $\left(1,\ 1\right)$  

(Type your answer with NO spaces!)

Find  $\frac{d^2y}{dx^2}$  for  $x=\cos\ y$  

•  $-\csc^2y\cot y$  

•  $-\csc y\cot y$  

•  $-\cot^2y$  

•  $-\sin x\sin y$