Implicit Differentiation

This is a simple guide about how to do implicit differentiation

Explained in a simple way, if there is a function that is not in the form

y = some expression with only x. There are many instances where y cannot be solved explicitly in terms of x, so implicit differentiation required. Other times, it is much faster and “cleaner” to use implicit differentiation

For example, given the equation x^2 + y^2 =49, implicit differentiation is the easiest way to find dy/dx

To do implicit differentiation, know that the derivative of y is dy/dx. In other words simply add a dy/dx after every y when the derivative is taken. Essentially, this is a continuation of the chain rule. Then, solve for dy/dx in terms of x and y

Steps

1. d/dx the entire equation
2. Differentiate x as normal
3. When differentiating y, add a dy/dx afterwards
4. Solve for dy/dx in terms of x and y

Example Problems

1. x^2 + y^2 = 49
1. d/dx(x^2 + y^2) = 49
2. 2x + 2y(dy/dx) = 0
3. Solve for dy/dx so dy/dx = -2x/2y
4. dy/dx = -x/y

2. xy=9 → This problem requires both product rule and implicit differentiation
1. d/dx(xy=9)
2. Use product rule: x*dy/dx + y*1 = 0
3. Solve for dy/dx
4. dy/dx = -y/x