- 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 ax
- 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
Introduction to :
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
Practice:
Example 1
If y is a function of x and x2 + y2 = 16, find y'. |
Example 2
Find y' given that y is a function of x and xy + x2 + y2 = 0. |
Exercise 1
Use implicit differentiation to find y', assuming in the case that y is a function of x.
- x3 + y3 = 4x
Exercise 2
Use implicit differentiation to find y', assuming in the case that y is a function of x.
- y = cos(y) + 2x
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.
- xy2 + x3y = 4x
Exercise 5
implicit differentiation to find y', assuming in the case that y is a function of x.
