Derivatives > Implicit Differentiation

**Implicit differentiation** is used to solve implicit expressions. Functions come in two flavors: explicit functions are in the form y = …. For example, y – 2x -5. This function could also be written as an *implicit* expression 2x – y = 5. While you could easily get this particular equation into an explicit form, sometimes it’s difficult, or impossible to solve for y;, which means you can’t use any of the “usual” methods of differentiation.

For example, the function 2xy = 1 can be easily solved for y, while a more complicated function, like 2y^{2} -cos y = x^{2} cannot.

With this technique, you directly differentiate both sides of the equation *without solving for y*. You’ll need to be comfortable with the chain rule, the power rule and various other techniques for finding derivatives.

## Implicit Differentiation Examples

**Example problem #1:** Differentiate x = e^{y} using implicit differentiation.

Step 1: **Rewrite the function**, placing dy/dx on both sides:

Step 2: **Differentiate the left side of the equation**. The derivative of x is 1, so:

Step 3: **Differentiate the right side of the equation**. This example requires the chain rule:

Step 4: **Use algebra to solve for y′**.

*That’s it!*

## Finding the Derivative Implicitly: Example #2

In the above example, I used Leibniz notation for the derivative. In this next example, I use prime notation. The two notations mean the same thing (the derivative), but—as you’ll see in this example, y′ is a bit easier to write out and work with.

**Example question #2**: Find the implicit derivative of x^{3} + 4y^{2} = 1.

Step 1: **Find the derivative(s) for the left hand side.**

For this example, there are two parts to find derivatives for: *x*^{3} and 4*y*^{2}:

(the power rule) =*x*^{3}

3*x***4***y*^{2}:- Differentiate with respect to
*y*, - Multiply by
*y*′.

Solution: 8

*yy*′.- Differentiate with respect to

Combine the two answers from above (The Sum Rule):

3*x* + 8*yy*′

Step 2: **Differentiate the right hand side.** The derivative of a constant is 0.

Step 3: **Put your answers** from the left hand side (Step 1) and right hand side (Step 2) **back together**:

3*x* + 8*yy*′ = 0

This is the implicit derivative.

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