Blog #6 - Mary Graci

This week we learned in Calculus how to take the derivative of an equation with two different variables, called implicit derivatives. First of all, when referring to the derivative of this kind of equation, you use d(insert other variable)/dx. If the equation is x + y = 2, then the derivative would be called dy/dx. If the equation is x – r = 6, then the derivative would be called dr/dx.

Ex. 1) x + y = 2
First, take the derivative of each individual term, including what’s after the equal sign. When taking the derivative of the second variable (y), write “dy/dx” after the derivative.
1 + dy/dx = 0
Now, solve for dy/dx.
dy/dx = -1
And that’s your derivative!

Ex. 2a) Take the implicit derivative of: x^2 – 3y^2 = 6
2x – 6y dy/dx = 0
-6y dy/dx = -2x
dy/dx = 2x/6y
dy/dx = x/3y

We also learned how to take a second implicit derivative. It’s basically the same concept as taking the second derivative of a normal equation. You take the derivative (as shown above), then take the derivative again, then plug in the first derivative for dy/dx, and solve. If when you are finished you can plug in your original equation into your second derivative to simplify the answer more, do that and that’s your answer.
*a second implicit derivative is represented by d^2y/dx^2

Ex. 2b) Now take the second implicit derivative of example 2.
Use the quotient rule.
d^2y/dx^2 = ((3y)(1) – (x)(3dy/dx)) / 9y^2
= (3y – 3x dy/dx) / 9y^2
= (y – x dy/dx) / 3y^2
= (y – x (x/3y)) / 3y^2
= ((3y^2/3y) – (x^2/3y)) / 3y^2
= ((3y^2 – x^2) / 3y) / 3y^2
Sandwich it.
d^2y/dx^2 = (3y^2 – x^2) / 9y^3
Be sure to simplify the answer as much as possible.