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Chapter 2: Q. 40 (page 233)

Find the derivatives of each of the functions in Exercises 17鈥�50. In some cases it may be convenient to do some preliminary algebra

f(x)=tan-1(lnx)

Short Answer

Expert verified

The derivative is1x1+(lnx)2.

Step by step solution

01

Step 1. Given Information.

The given function isf(x)=tan-1(lnx).

02

Step 2. Preliminary Algebra.

We know,

ChainRule,(f鈭�u)'(x)=f'(u(x))u'(x)Also,ddxlnx=1x

03

Step 3. Derivative of the function.

The derivative of the function,

ddxtan-1(lnx)=11+(lnx)2ddx(lnx)=11+(lnx)21x=1x1+(lnx)2

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Most popular questions from this chapter

Use the definition of the derivative to find f'for each function fin Exercises 34-59

role="math" localid="1648284617718" f(x)=2x+1

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

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.

Stuart left his house at noon and walked north on Pine Street for 20minutes. At that point he realized he was late for an appointment at the dentist, whose office was located south of Stuart鈥檚 house on Pine Street; fearing he would be late, Stuart sprinted south on Pine Street, past his house, and on to the dentist鈥檚 office. When he got there, he found the office closed for lunch; he was 10minutes early for his 12:40appointment. Stuart waited at the office for 10minutes and then found out that his appointment was actually for the next day, so he walked back to his house. Sketch a graph that describes Stuart鈥檚 position over time. Then sketch a graph that describes Stuart鈥檚 velocity over time.

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
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