## 1,001 Calculus Practice Problems

__Part I__

__Part I__

**The Questions**

__Chapter 7__

__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.** *x*^{2} + *y*^{2} = 9

**409.** *y*^{5} + *x*^{2}*y*^{3} = 2 + *x*^{2}*y*

**410.** *x*^{3}*y*^{3} + *x* cos(*y*) = 7

**411.**

**412.**

**413.**

*Using Implicit Differentiation to Find a Second Derivative*

*414–417** Use implicit differentiation to find **.*

**414.** 8 *x*^{2} + *y*^{2} = 8

**415.** *x*^{5} + *y*^{5} = 1

**416.** *x*^{3} + *y*^{3} = 5

**417.**

*Finding Equations of Tangent Lines Using Implicit Differentiation*

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

**418.** *x*^{2} + *xy* + *y*^{2} = 3 at (1, 1)

**419.** at (2, 1)

**420.** *x*^{2} + 2*xy* + *y*^{2} = 1 at (0, 1)

**421.** cos(*xy*) + *x*^{2} = sin *y* at

**422.** *y*^{2}(*y*^{2} – 1) = *x*^{2 }tan *y* at (0, 1)