Science:Math Exam Resources/Courses/MATH100/December 2011/Question 01 (h)/Solution 1

In order to find the slope of the tangent line, we need to take the derivative. However, because this expression is not written in the form y = f(x), we will have to use implicit differentiation with respect to x.

Thus we implicitly differentiate the expression with respect to x. Note that the term x² can be differentiated directly, the term xy requires the product rule, and the term y² requires the chain rule.

$2x+y+x{\frac {dy}{dx}}+2y{\frac {dy}{dx}}=0$

Since we are finding a specific derivative at the point (0,2), we can go ahead and plug in the values x=0 and y=2 to get:

$2(0)+(2)+(0){\frac {dy}{dx}}+2(2){\frac {dy}{dx}}=0$

$2+4{\frac {dy}{dx}}=0$

If we solve the above expression for the derivative dy/dx we get:

${\frac {dy}{dx}}={\frac {-2}{4}}={\frac {-1}{2}}$

So the slope of the tangent line is -1/2.