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Chapter 2: Q. 53 (page 184)

Use the definition of the derivative to find ffor each function fin Exercises 39-54

f(x)=x3x+1

Short Answer

Expert verified

The derivative isf'(x)=2x3+3x2(x+1)2

Step by step solution

01

Step 1.Given information

Given functionf(x)=x3x+1

02

Use the chain rule and calculate

Calculating, we get

f(x)=x3x+1f'(x)=(x+1).3x2-x3.1(x+1)2f'(x)=3x3+3x2-x3(x+1)2f'(x)=2x3+3x2(x+1)2

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Most popular questions from this chapter

Differentiation review: Without using the chain rule find the derivative of each of the function f that follows some algebra may be required before differentiating

a)f(x)=(3x+1)4b)f(x)=(1x+1x2)2c)f(x)=(x+1)2xd)f(x)=(x+1)2x

For each function fgraphed in Exercises 65-68, determine the values of xat which ffails to be continuous and/or differentiable. At such points, determine any left or right continuity or differentiability. Sketch secant lines supporting your answers.

Taking the limit: We have seen that if f is a smooth function, then f'(c)≈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→0f(c+h)-f(c)h.

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

The total yearly expenditures by public colleges and universities from 1990 to 2000 can be modeled by the function E(t)=123(1.025)t, where expenditures are measured in billions of dollars and time is measured in years since 1990.

(a) Estimate the total yearly expenditures by these colleges and universities in 1995.

(b) Compute the average rate of change in yearly expenditures between 1990 and 2000.

(c) Compute the average rate of change in yearly expenditures between 1995 and 1996.

(d) Estimate the rate at which yearly expenditures of public colleges and universities were increasing in 1995.

Prove, in two ways, that the power rule holds for negative integer powers

a) by using the z→xdefinition of the derivative

b) by using theh→0definition of the derivative
See all solutions

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