IMPLICIT Differentiation

Total Recall We write the derivative of a function as: In general, the means to take the derivative with respect to “x.” We will now work with functions where this is NOT exactly possible.

Something Different How would be computed for Since this curve is NOT a function, the derivative still makes sense because the slope of the tangent line will be on the curve When a curve is expressed in terms the independent & dependent variable, it is an IMPLICIT function.

The METHOD Step1: Step 2: Step 3: Step 4: take the derivative of each side of the equation implicitly Step 2: Rearrange the equation so the terms containing are on one side and the other terms are on the other side Step 3: Factor out , and then Step 4: SOLVE for

EXAMPLE Find the derivative:

EXAMPLE Find the derivative: In this example, the derivative of y with respect to x is

EXAMPLE Find the derivative:

EXAMPLE Find the derivative: Notice the derivative of y2 has 2y and

EXAMPLE So in our ORIGINAL example:

EXAMPLE Find the derivative:

EXAMPLE Find the derivative:

EXAMPLE Find the derivative:

SUMMARY Remember, just like explicitly defined functions, the derivative may NOT always exist. If the graph of a curve has a corner, cusp, or vertical tangent line at a point, then it does not make sense to compute a derivative Remember, the derivative is the SLOPE of the tangent like to a curve