Implicit differentiation is the process of computing the derivative of an implicitly defined function without first finding an explicit representation for the function.
 

Derivatives of inverses One context in which this comes up naturally is the task of finding the derivative of the inverse of a function. It's not uncommon to know that a function has an inverse without being able to find an explicit formula for that inverse.

For example, consider the function f with the formula f(x) = x⁵ + 3x³ + 4x.

The derivative of f has the formula f'(x) = 5x⁴ + 9x² + 4, so we know that f'(x) is positive for every value of x. This means that f is always increasing, so f must have an inverse.

That doesn't mean that f has an inverse with a formula, however.

Nonetheless, we can find a formula for the derivative of the inverse of f.
 

Suppose that y = f^-1(x), where f has the formula above, and the we want a formula for y' .

Since y = f^-1(x), x = f(y), so x = y⁵ + 3y³ + 4y. We know this formula makes y a function of x, so we can use implicit differentiation to find a formula for y'.

Do you recognize the formula in the denominator of your answer?

Can you get a derivative for y which doesn't have y's in it?

This process works fairly well for finding derivatives of the arcsin, arctan, and ln functions. Of course, the calculator already knows derivatives for these functions, so this information is of limited practical value.
 

Exercises:

Find a formula for the derivative of the inverse of the function f (assume it has one), where f(x) is given by the following formulas. Your formulas might have to include f^-1(x), but see if you can simplify enough to avoid that.
 

• f(x) = x³ + x + 1
• f(x) = e ^x
• f(x) = e ^2x+3
• f(x) = tan(x)
• f(x) = ln(x)
• f(x) = sin(x)