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Chapter 2: Q. 89 (page 200)

use the definition of the derivative to prove the quotient rule

Short Answer

Expert verified

We use the definition of derivative to prove the quotient rule

Step by step solution

01

Given information

We are given a function of the formf(x)g(x)

02

use the definition to find the derivative

We get,

limh鈫�0f(x+h)g(x+h)-f(x)g(x)hlimh鈫�0f(x+h)g(x)-g(x+h)f(x)g(x+h)g(x)hlimh鈫�0f(x+h)g(x)-f(x)g(x)+f(x)g(x)-g(x+h)f(x)g(x+h)g(x)hlimh鈫�0g(x)[f(x+h)-f(x)]-f(x)[g(x+h)-g(x)]g(x+h)g(x)hlimh鈫�0g(x)f(x+h)-f(x)h-f(x)g(x+h)-g(x)hg(x+h)g(x)=g(x)f'(x)-f(x)g'(x)[g(x)]2

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Most popular questions from this chapter

For each function f(x)and interval localid="1648297458718" a,bin Exercises localid="1648297462718" 81-86, use the Intermediate Value Theorem to argue that the function must have at least one real root on localid="1648297466951" a,b. Then apply Newton鈥檚 method to approximate that root.

localid="1648297471865" f(x)=x3-3x+1,a,b=1,2

Each graph in Exercises 31鈥�34 can be thought of as the associated slope function f' for some unknown function f. In each case sketch a possible graph of f.

Use the definition of the derivative to find f'for each function fin Exercises39-54

f(x)=21-x

Use the definition of the derivative to find f'for each function fin Exercises 39-54

role="math" localid="1648290170541" f(x)=x-1x+3

Taking the limit: We have seen that if f is a smooth function, then f'(c)鈮�f(c+h)-f(c)hThis approximation should get better as h gets closer to zero. In fact, in the next section we will define the derivative in terms of such a limit.

f'(c)=limh鈫�0f(c+h)-f(c)h.

Use the limit just defined to calculate the exact slope of the tangent line tof(x)=x2atx=4.
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