Handout Example Section 3 - Radford University



Handout Example Section 3.6 Implicit Differentiation

Example: Find the equation of the tangent line to the curve [pic] at the point [pic].

Solution: To find the equation of the tangent line, we need its slope and a point on the line. The point [pic] is given so we first must find the tangent line slope. Recall that the slope of the tangent line is found by finding the derivative [pic], which in this example we must find by implicit differentiation. Thus, we have

[pic]

[pic] (Use fact that [pic] to rewrite in an easier

form to differentiate )

[pic] (Differentiate both sides)

[pic] (Simplify)

[pic] (Isolate [pic] term)

[pic] (To solve for [pic], multiply both sides by 18)

[pic] (Simplify fractions in previous step)

[pic] (Divide both sides by y to solve for [pic])

To find the tangent line slope, we must substitute the point [pic] into [pic]. Note here that both the x and y coordinates of the point are required. Thus, we obtain

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[pic].

Using the slope intercept equation of a line [pic], we now write the equation of the line as follows:

[pic]

Thus, the tangent line equation is [pic].

[pic]

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