1,001 Calculus Practice Problems

Part I

The Questions

 

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 9781118496718-eq07001.eps.

408. x2 + y2 = 9

409. y5 + x2y3 = 2 + x2y

410. x3y3 + x cos(y) = 7

411. 9781118496718-eq07002.eps

412. 9781118496718-eq07003.eps

413. 9781118496718-eq07004.eps

Using Implicit Differentiation to Find a Second Derivative

414–417 Use implicit differentiation to find 9781118496718-eq07005.eps.

414. 8 x2 + y2 = 8

415. x5 + y5 = 1

416. x3 + y3 = 5

417. 9781118496718-eq07006.eps

Finding Equations of Tangent Lines Using Implicit Differentiation

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

418. x2 + xy + y2 = 3 at (1, 1)

419. 9781118496718-eq07007.eps at (2, 1)

420. x2 + 2xy + y2 = 1 at (0, 1)

421. cos(xy) + x2 = sin y at 9781118496718-eq07008.eps

422. y2(y2 – 1) = xtan y at (0, 1)