- 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

Things to remember for implicit differentiation:

*x'*= 1

- since
*y*is a function of*x*, any derivative involving*y*must use the chain rule

- since
*y*is a function of*x*, taking the derivative of*xy*(or of any other product involving both*x*and*y*) requires the product rule

- since
*y*is a function of*x*, taking the derivative of (or any other quotient involving both*x*and*y*) requires the quotient rule

Example 1

If
find |

Example 2

Find |

Exercise 1

Use implicit differentiation to find *y'*, assuming in the case that *y* is a function of *x*.

*x*^{3}+*y*^{3}= 4*x*

Exercise 2

Use implicit differentiation to find *y'*, assuming in the case that *y* is a function of *x*.

*y*= cos(*y*) + 2*x*

Exercise 3

implicit differentiation to find *y'*, assuming in the case that *y* is a function of *x*.

- e
^{{y2}}-*x*=*y*

Exercise 4

implicit differentiation to find *y'*, assuming in the case that *y* is a function of *x*.

*xy*^{2}+*x*^{3}*y*= 4*x*

Exercise 5

implicit differentiation to find *y'*, assuming in the case that *y* is a function of *x*.