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Computing Derivatives

Computing Derivatives

At a Glance - Using Lagrange Notation

Not a fan of Leibniz notation? We can do implicit differentiation with Lagrange notation just as well.

Things to remember for implicit differentiation with Lagrange notation:

  • 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 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 that y is a function of x.

  • x3 + y3 = 4x

Exercise 2

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

  • y = cos(y) + 2x

Exercise 3

What is y' in the following equation?

  •  ey2x = y

Exercise 4

Using implicit differentiation, what is y ' in the following equation?

  • xy2 + x3y = 4x

Exercise 5

What is y ' in the following equation?


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