- Topics At a Glance
- Derivatives of Basic Functions
- Constant Functions
- Lines
- Power Functions
- Exponential Functions
- Logarithmic Functions
- Trigonometric Functions
- Derivatives of More Complicated Functions
- Derivative of a Constant Multiple of a Function
- Multiplication by -1
- Fractions With a Constant Denominator
- More Derivatives of Logarithms
- Derivative of a Sum (or Difference) of Functions
- Derivative of a Product of Functions
- Derivative of a Quotient of Functions
- Derivatives of Those Other Trig Functions
- Solving Derivatives
- Using the Correct Rule(s)
- Giving the Correct Answers
- Derivatives of Even More Complicated Functions
- The Chain Rule
- Re-Constructing the Quotient Rule
- Derivative of a
^{x} - Derivative of
*ln*x - Derivatives of Inverse Trigonometric Functions
- The Chain Rule in Leibniz Notation
- Patterns
- Thinking Backwards
**Implicit Differentiation**- Computing Derivatives Using Implicit Differentiation
- Using Leibniz Notation
- Using Lagrange Notation
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

Now it's time to throw a monkey wrench into the works, curve ball style. What happens when we mix two variables together, on both sides of the equation? Why calculus? Why?! We're entering the Twilight Zone, we're traveling to another dimension. Once the room stops spinning, we'll move on. Good now? Good. We can handle this.

A function is a special kind of relation where each input has exactly one output. Often we have a rule or a formula that tells us how to reach the input to the output, such as

*y* = *x* + 5.

In such cases we say the function is defined **explicitly**. The dependent variable appears all by itself on one side of the equation, and the other side gives a recipe for making the dependent variable from the independent variable.

We say a function is defined **implicitly** if we have an equation in which the dependent variable doesn't have a side of the equation all to itself. The equation

5*x*^{2} - 4*y* = 7

is an implicit definition of *y*, since it doesn't actually say how to go from *x* to *y*. If we solve this equation for *y* we find the explicit definition

We can say that *y* is "defined implicitly" even if *y* isn't actually a function. For example, the equation

*x*^{2} + *y*^{2} = 1

describes a circle of radius 1. A circle isn't a function, since it fails the vertical line test. However, we would still say that this equation implicitly defines *y*.

For a total headache, we might even say "*y* is an implicitly defined function" even though *y* isn't a function. We probably worry about this distinction in class, but we wanted to point it out for the sake of completeness!

Exercise 1

Determine whether *y* is defined explicitly or implicitly.

*y*= 4*x*+ 3*y*

Exercise 2

Determine whether *y* is defined explicitly or implicitly.

*y*= 7*x*+*x*^{2}

Exercise 3

Determine whether *y* is defined explicitly or implicitly.

*x*^{4}+*y*= 6*y*

Exercise 4

Determine whether *y* is defined explicitly or implicitly.

Exercise 5

Determine whether *y* is defined explicitly or implicitly.

- 12
*x*+ 16*y*=*x*^{2}-*y*- 8