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The trouble with isolating y...

The trouble with isolating y. Is the relation y = x 3 + x a function? Justify your answer. Is the inverse of the above relation a function? Again, justify!. Graph:. Equation:. (Erase). Is y a function of x ?. Can we isolate the dependent variable?.

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The trouble with isolating y...

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  1. The trouble with isolating y... Is the relation y = x3 + x a function? Justify your answer. Is the inverse of the above relation a function? Again, justify! Graph: Equation: (Erase) Is y a function of x? Can we isolate the dependent variable?

  2. As you have seen on the previous slide, sometimes isolating the dependent variable of a function is difficult or even impossible. However, this should not prevent us from finding the slope of the function! (i.e. determining the derivative) Original equation Take the derivative of both sides Take the derivative of each term separately Use chain rule to take the derivative of y2 Common factoring Isolating the derivative

  3. Practice: Determine the slope of the function 2xy - y3 = 4 at the point (3, 2).

  4. Note: Implicit differentiation may be useful even in cases where it is possible to isolate the dependent variable For example, calculate the slope of the circle x2 + y2 = 25 at (3, -4): Explicit differentiation: Implicit differentiation: Is there an advantage to using the second method to solve this problem?

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