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What do you do when you can’t separate out y? 2.5 Implicit Differentiation
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Up to now we have seen most equations in explicit form – that is, y in terms of x, like y = 2x 6 -5 (solved for y alone) Now we will work with equations written implicitly, like xy=8. This is the implicit form; it can be rewritten explicitly as y = 8/x Sometimes we can isolate y. Sometimes we can’t! For example, x 2 + 2y 3 + 4y = 2. So we will learn how to implicitly differentiate to handle any situation. When we find dy/dx, we are differentiating with respect to x. Anytime we see a term with x alone, we differentiate as usual. Whenever we differentiate a term involving y, we must apply the Chain Rule, because you are assuming there is some function where y could be written implicitly.
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When we find dy/dx, we are differentiating with respect to x. Anytime we see a term with x alone, we differentiate as usual. Whenever we differentiate a term involving y, we must apply the Chain Rule, because you are assuming there is some function where y could be written implicitly. Ex 1 p. 141 Differentiating with Respect to x Variables agree: Use simple power rule Variables disagree: Use Chain Rule Product Rule,Chain
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<ul><li>Guidelines for Implicit Differentiation </li></ul><ul><li>Differentiate both sides of the equation with respect to x. </li></ul><ul><li>Collect all terms involving dy/dx on the left side of the equation and move all other terms to the right side of the equation. </li></ul><ul><li>Factor dy/dx out of the left side of equation </li></ul><ul><li>Solve for dy/dx </li></ul><ul><li>Results can be a function in both x and y </li></ul>
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<ul><li>Ex 2 p. 142 Find dy/dx given that y 3 + y 2 – 5y – x 2 = -4 </li></ul><ul><li>Solution </li></ul><ul><li>Differentiate both sides of equation with respect to x. </li></ul><ul><li>Collect dy/dx terms on one side, rest on other </li></ul><ul><li>Factor out dy/dx </li></ul><ul><li>4. Solve for dy/dx </li></ul>
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Input y = t, y=t for Play around with window values until you get graph shown. Would you like to see what this graph looks like?
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Ex 3 p. 143 Representing graphs by differentiable functions. If possible, represent y as a differentiable function of x. Just a single point, so not differentiable Differentiable except at (1, 0) and (-1, 0) Differentiable except at (1, 0)
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Ex 4 p.143 Finding the Slope of a Graph Implicitly Determine the slope of the tangent line to the graph of Differentiate with respect to x Get dy/dx terms alone, then solve for dy/dx Substitute x and y from point of tangency and simplify. If you want to do it the hard way, solve original equation for y and differentiate that.
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Ex 5 p. 144 Finding the Slope of a Graph Implicitly Determine the slope of Plug in point (3, 1) This graph is called a lemniscate
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Ex 6, p144 Determining a Differentiable Function Range Find dy/dx implicitly for equation sin y = x (note: inverse function y = sin -1 x). Find the largest interval for y values on which x is differentiable. Or alternatively, Graph becomes vertical at endpoints of interval!
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