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Implicit derivatives

If it is not possible to express y explicitly as a function of x in an equation, then you must differentiate implicitly. You write it with the differential operator as

where this symbol stands for "take the derivative of...".



The chain rule is used extensively in the form

to determin the derivatives of the terms with y in it. Sometimes it's simply easier to differentiate a function implicit, rather than explicit.


Example 1

The equation f (x) = xx can not easily be differentiated, because both the base and the exponent are variable. By first taking the logarithm, we eliminate the exponent

what we convert to

Now we implicitly differentiate both sides to x

The left side can be calculated using the chain rule

The derivative of the logarithm and the product rule gives

Multiplying by y gives

Set again y = xx, then the solution is


Example 2

The circle with radius r is given by the equation x2 + y2 = r2. Implicit derivation gives

It follows that the tangent to the circle at the point (x, y) has the slope .


Example 3

The implicit derivative of the function x y – 3x – 2y + 5 = 0 gives



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