009A Sample Final A, Problem 4

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4. Find an equation for the tangent line to the function   at the point .

Since only two variables are present, we are going to differentiate everything with respect to in order to find an expression for the slope, . Then we can use the point-slope equation form at the point to find the equation of the tangent line.
Note that implicit differentiation will require the product rule and the chain rule. In particular, differentiating can be treated as
which has as a derivative


Finding the slope:  
We use implicit differentiation on our original equation to find
From here, I would immediately plug in to find :
     , or
Writing the Equation of the Tangent Line:  
Now, we simply plug our values of and  into the point-slope form to find the tangent line through is
or in slope-intercept form

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