Implicit Differentiation

Implicit Differentiation

Implicit differentiation is just a method for finding a derivative when

is not expressed explicitly as a function of . We use the Chain Rule.

Example: Find if

Solution: We differentiate both sides with respect to

.

then moves terms around to solve for

To find a second derivative we differentiate with respect to

again, in this case, using the quotient rule.

and substitute the quantity we found earlier

and substitute from the original equation

This last bit doesn't always turn out so nicely but it frequently does with standard functions.

STEP 2011, Math I, #1: (i) Show that the gradient of the curve

where , is

lies on both the straight line and the curve , where . Given that, at this point, the line

and the curve have the same gradient, show that

.

, where , is a normal to the curve , then .

STEP 2006, Math II, #7: An ellipse has equation

. Show that the equation of the tangent at the point is

The point A has coordinates

, where and are positive. The point E has coordinates and the point P has coordinates

where . The line through E parallel to AP meets the line at the point Q. Show that the line PQ is tangent to the

above ellipse at the point given by

.

Determine by means of sketches, or otherwise, whether this result holds also for

and , show that the -coordinate of the points where the tangent meets the curve satisfy

.

Hence find positive integers

, , and such that

of a curve at a point satisfies