Can anyone tell me how I have done with these ones on Implicit & Exponential Differentiation?
a)
$\displaystyle x^2 + y^2 - 2x - 6y + 5 = 0$
Now we differentiate it implicitly:
$\displaystyle 2x + 2yy' - 2 - 6y' = 0$
$\displaystyle 2yy' - 6y' = -2x + 2$
$\displaystyle y'(2y - 6) = 2x + 2$
$\displaystyle y' = \frac{2x + 2}{2y - 6}$
b)
$\displaystyle x^2 + 2xy + 3y^2 = 4$
Implicitly:
$\displaystyle 2x + (2y + 2xy') + 6yy' = 0$
$\displaystyle 2x + 2y + 2xy' + 6yy' = 0$
$\displaystyle 2xy' + 6yy' = -2x - 2y$
$\displaystyle y'(2x + 6y) = -2x - 2y$
$\displaystyle y' = \frac{-x - y}{x + 3y}$
With this you are forgetting that when you differentiate y in terms of x then you must do something along the lines of:
$\displaystyle \frac{dy}{dx}[y^2] = 2y\frac{dy}{dx}$
$\displaystyle =2yy'$