AP Calculus

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AP Calculus

• Implicit Differentiation

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This is How We Do It.

• Consider Y vX..
• dy/dx ½ x-1/2
• This function is called an explicit function
  because y is clearly defined in terms of x.
• BUT

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What if

• Y wasnt clearly defined in terms of x?
• Using our previous example, lets say y2x. How
  would you find dy/dx?
• You could solve for Y and take the derivative
  directly or you could be sneaky and take the
  derivative implicitly. (Hmmmm. I dont like the
  sound of that word.)

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This is going to be so easy!!!!

• Lets pretend y is some function of x which we
  dont know. This is similar to the chain rule
  when we let u be some function of x. Just like
  the chain rule we will take the derivative of y2
• It would be the derivative of the outside times
  the derivative of the inside

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More Math Magic

• Y2x
• Step 1 Take the derivative of left side and of
  the right side with respect to x.
• 2y(y) 1
• Step 2 Solve for y
• Y 1/2y
• Step 3 Substitute what y equals into equation
• Y 1/(2vx)

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So what.

• Implicit differentiation allows you take the
  derivative of a function even if you dont know
  the exact function and once that function is
  known you can express the derivative in terms of
  x.