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Chapter 7

Implicit Differentiation

When 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.

Using Implicit Differentiation to Find a Derivative

408–413 Use implicit differentiation to find .

408.

409.

410.

411.

412.

413.

Using Implicit Differentiation to Find a Second Derivative

414–417 Use implicit differentiation to find .

414.

415.

416.

417.

Finding Equations of Tangent Lines Using Implicit Differentiation

418–422 Find the equation of the tangent line at the indicated point.

418. at

419. at

420. at

421. at

422. at