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Chapter 2: Q. 71 (page 211)

Each of the equations in Exercises 69鈥�80 defines y as an implicit function of x. Use implicit differentiation (without solving for y first) to find dydx

xy2+3x2=4

Short Answer

Expert verified

dydx=-y2-6x2xy

Step by step solution

01

Step 1. Given Information:

Given equation: xy2+3x2=4

We want to finddydxdefines y as an implicit function of x by use implicit differentiation.

02

Step 2. Solution:

Differentiate both sides w.r.t. x

ddx(xy2+3x2)=ddx(4)ddx(xy2)+ddx(3x2)=ddx(4)

Using Product and Chain Rule we get

(xddxy2+y2ddxx)+6x=0x路2y路dydx+y2+6x=02xy路dydx=-y2-6xdydx=-y2-6x2xy

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Most popular questions from this chapter

In Exercises 69鈥�80, determine whether or not fis continuous and/or differentiable at the given value of x. If not, determine any left or right continuity or differentiability. For the last four functions, use graphs instead of the definition of the derivative.

fx=x2,ifx鈮�12x-1,ifx>1,x=1

Use (a) the h鈫�0definition of the derivative and then

(b) the z鈫�cdefinition of the derivative to find f'(c) for each function f and value x=c in Exercises 23鈥�38.

28.f(x)=x4+1,x=2

For each function f and value x=cin Exercises 35鈥�44, use a sequence of approximations to estimate f'(c). Illustrate your work with an appropriate sequence of graphs of secant lines.

f(x)=4-x2,c=1

The following reciprocal rules tells us hoe to differentiate the reciprocal of a function

ddx(1f(x))=-1[f(x)]2

Prove this using

a) definition of the derivative

b) by using the quotient rule

If Katie walked at 3miles per hour for 20minutes and then sprinted at 10miles an hour for 8minutes, how fast would Dave have to walk or run to go the same distance as Katie did at the same time while moving at a constant speed? Sketch a graph of Katie鈥檚 position over time and a graph of Dave鈥檚 position over time on the same set of axes.
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