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Implicit derivative
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...".
Explanation
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 function 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
so