Implicit Differentiation
Finding the slope when equations won't let you isolate y
What You Need to Know
The Problem
Some equations mix x and y so thoroughly you can't rearrange them into y = f(x)
The Solution
Differentiate both sides with respect to x, treating y as a function of x
Key Tool
The chain rule: whenever you differentiate a y-term, multiply by dy/dx
The Rearrangement
After differentiating, collect all dy/dx terms on one side and solve
1. Why Implicit Differentiation?
In most calculus problems, you're given equations in the form y = f(x)—these are called explicit equations. You just differentiate the right side and you're done.
But some equations are messy. Take:
sin y = 3x²e^(-4y)
Try to rearrange this to get y alone. You can't. The x and y are completely tangled together. This is an implicit equation—y is hidden implicitly in the relationship, rather than explicitly isolated.
But we still need to find dy/dx. That's where implicit differentiation comes in: we differentiate both sides of the equation with respect to x, and the chain rule does the heavy lifting.
Key insight: You don't need to solve for y. You don't even need y explicitly. You just need an expression for dy/dx in terms of both x and y, and that's useful enough.
2. The Chain Rule for y-Functions
The foundation of implicit differentiation is this: whenever you differentiate a function of y with respect to x, you must multiply by dy/dx.
Let's say f(y) is some function of y, like sin(y) or e^y or y³.
Why?
Because y itself depends on x. By the chain rule:
d/dx[f(y)] = df/dy · dy/dx
When you differentiate a y-function, you get the derivative with respect to y, then you multiply by dy/dx (the rate of change of y with respect to x).
Example: sin(y)
The derivative of sin(y) with respect to y is cos(y). So:
d/dx[sin y] = cos y · dy/dx
Remember: Every y-term gets a dy/dx tag attached when you differentiate. X-terms don't—they're independent variables.
3. Product Rule + Chain Rule = Power
When a term involves both x and y multiplied together, the product rule and chain rule work together beautifully.
In words: Differentiate the first function (x-function) and leave the second alone. Then leave the first and differentiate the second (remembering the dy/dx). Add them together.
Example: 3x² · e^(-4y)
Here f(x) = 3x² and g(y) = e^(-4y).
d/dx[3x² · e^(-4y)] = 6x · e^(-4y) + 3x² · (-4e^(-4y)) · dy/dx
= 6x·e^(-4y) - 12x²·e^(-4y) · dy/dx
Common pattern: When you have a term like xy or x²y or x·sin(y), you always use the product rule, and the y-part gets a dy/dx attached.
4. Four Essential Formulas (Memorise These)
These four formulas capture the most common patterns you'll see. Learn them once, use them forever.
Formula 1: Chain Rule for y-Functions
Special case (power rule): If you have y³ or y^(1/2) or any power of y, just apply the power rule and attach dy/dx.
Examples: