Question

F(y, x)=x³-2 x² y+3 xy²-22=0 , is an implicit function y=f(x) defined around the point (y=3, x=1) If your answer is affirmative, evaluate dy/dx at

Answer 1

The question is about using implicit differentiation to find dy/dx for the function given by F(y, x)=x³-2 x² y+3 xy²-22=0 at the point (y=3, x=1).

Step-by-step explanation:

To determine whether F(y, x)=x³-2 x² y+3 xy²-22=0 is an implicit function y=f(x) defined around the point (y=3, x=1), we need to apply the implicit differentiation principle. At the point (y=3, x=1), we substitute the values into the equation to see if they satisfy it.

Next, we differentiate the equation with respect to x, treating y as a function of x (y=f(x)). This gives us:
3x² - 4xy - 2x² (dy/dx) + 3y² + 6xy (dy/dx) = 0

Now we solve for dy/dx to find the derivative at the point given. Once we have dy/dx, we can plug in x=1 and y=3 to get the value of the derivative at that point. The steps involve rearranging the terms, factoring out dy/dx, and then solving for dy/dx.