Implicit Differentiation

Suppose the equation defines the relation between variables and . Thinking of as the independent variable, we apply the chain rule to differentiate the equation and then solve the resulting equation for the derivative . This process is called implicit differentiation. The process results in the equation

Exercise: Find the equation of the tangent line to the graph of the equation at the given points:

(a) (b)

(c) (d)

(e) (f)

(g) (h)

(i) (j)

Higher-Order Derivatives of an Implicit Function

If satisfies the equation , then . This is the result of the chain rule . To find the second derivative , apply the formula

i.e., replace and in by . To find the third derivative , replace in by and by

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