Implicit Differentiation is a technique used to differentiate a function when it cannot be explicitly solved for the dependant variable. We will consider two methods for implicitly differentiating a function. The first method is taught to students taking Calculus 1, however the second method requires knowledge of partial derivatives and is not always taught in Multivariable Calculus classes.
For instance consider the function:
y2 + 3y – x2 = 3x
The derivative of y with respect to x (dy/dx) cannot be found by conventional means because the function cannot be explicitly solved for y. Therefore the function must be differentiated implicitly.
The basic method for implicit differentiation involves differentiating both sides of the equation with respect to x while treating y as being implicitly defined as a function of x.
For the function:
y2 + 3y – x2 = 3x
Differentiating implicitly yields:
2y(dy/dx) + 3(dy/dx) - 2x = 3
When terms involving y are differentiated the Chain Rule must be applied because it is assumed that y is implicitly defined as a function of x. Thus terms involving y will always have a dy/dx term attached.
Factoring out the dy/dx yields:
dy/dx(2y + 3) = 2x + 3
The final answer, the derivative of y with respect to x:
dy/dx = (2x + 3)/ (2y + 3)
Lets review the general guidelines for implicitly differentiating a function. First differentiate both sides of the function with respect to x or whatever the independent variable may be. Then collect all the terms involving dy/dx (the derivative) on one side of the equation and all other terms on the opposite side of the equation. Factor out the dy/dx term, and finally solve for dy/dx.
We will now consider another method for implicit differentiating a function.
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