At a Glance - Using Leibniz Notation
Some things to remember for implicit differentiation:
- 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.
With these things in mind, we're ready to get cracking.
Example 1
Find given that x^{2} + y^{2} = 4. |
Example 2
What is the derivative of y with respect to x given that 4y^{2} + 8y = 2x^{2}? |
Example 3
Assuming y is a function of x and xy + x = y, what is ? |
Exercise 1
Use implicit differentiation to find , assuming that y is a function of x.
- 4^{y} = x
Exercise 2
In the following equation what is the derivative of y with respect to x?
- sin y = x^{2}
Exercise 3
Find the derivative of y with respect to x in the following equation
- ln (xy) = x + y
Exercise 4
Find in the following equation.
- y^{3} + x^{2}y – x = y
Exercise 5
What is in the following equation?