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Use logarithmic differentiation to find the derivative of given function

f(x)=(sinx)cosx

Short Answer

Expert verified

The derivative of given function is

f(x)=sinxln(sinx)+cos2xsinx(sinx)cosx

Step by step solution

01

Step 1.Given data 

We have been given a function to find the derivative for them

f(x)=(sinx)cosx

02

Step 2.Taking log on both side to get the differentiation 

f(x)=(sinx)cosxlnf(x)=ln(sinx)cosx[Taking log in both sides]lnf(x)=cosxln(sinx)f(x)f(x)=ddx(cosxln(sinx))Differentiate both sideswith respect toxf(x)f(x)=ddx(cosx)ln(sinx)+cosxddxln(sinx)

=sinxln(sinx)+cosxsinxddxsinx=sinxln(sinx)+cosxsinx(cosx)

Using product rule and chain rule

f(x)=sinxln(sinx)+cosxsinx(cosx)f(x)

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Most popular questions from this chapter

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

A tomato plant given xounces of fertilizer will successfully bear T(x)pounds of tomatoes in a growing season.

(a) In real-world terms, what does T(5)represent and what are its units? What does T'(5)represent and what are its units?

(b) A study has shown that this fertilizer encourages tomato production when less than 20ounces are used, but inhibits production when more than 20ounces are used. When is T(x)positive and when is T(x)negative? When is T'(x)positive and when is T'(x)negative?

In Exercises 69–80, 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,ifx12x-1,ifx>1,x=1

Last night Phil went jogging along Main Street. His distance from the post office t minutes after 6:00p.m. is shown in the preceding graph at the right.

(a) Give a narrative (that matches the graph) of what Phil did on his jog.

(b) Sketch a graph that represents Phil’s instantaneous velocity t minutes after 6:00p.m. Make sure you label the tick marks on the vertical axis as accurately as you can.

(c) When was Phil jogging the fastest? The slowest? When was he the farthest away from the post office? The closest to the post office?

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

a) by using the zxdefinition of the derivative

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