Chapter 7
Implicit Differentiation
W
hen you know the techniques of implicit differentiation (this chapter) and
logarithmic differentiation (covered in Chapter 6), you’re in a position to find the
derivative of just about any function you encounter in a single-variable calculus course.
Of course, you’ll still use the power, product, quotient, and chain rules (Chapters 4 and 5)
when finding derivatives.
The Problems You’ll Work On
In this chapter, you use implicit differentiation to
✓ Find the first derivative and second derivative of an implicit function
✓ Find slopes of tangent lines at given points
✓ Find equations of tangent lines at given points
What to Watch Out For
Lots of numbers and variables are floating around in these examples, so don’t lose
your way:
✓ Don’t forget to multiply by dy/dx at the appropriate moment! If you aren’t getting the
correct solution, look for this mistake.
✓ After finding the second derivative of an implicitly defined function, substitute in the
first derivative in order to write the second derivative in terms of x and y.
✓ When you substitute the first derivative into the second derivative, be prepared to
further simplify.