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

f(x)=ln(xsinx).

Short Answer

Expert verified

The derivative of the given function is:cotx+1x.

Step by step solution

01

Step 1. Given Information.

f(x)=ln(xsinx).

02

Step 2. General formulas for finding derivatives.   

ddxxn=nxn-1,ddxsinx=cosx,ddxcosx=-sinx,ddxtanx=sec2x,ddx(secx)=secxtanx,ddx(cscx)=-cscxcotx,ddx(cotx)=-csc2x,Productrule:ddx(uv)=udvdx+vdudx,Quotientrule:ddxuv=vdudx-udvdxv2,Chainrule:ddxfog(x)=ddxf(g(x))×ddx(g(x).

03

Step 3. Solving derivative using above formulas. 

f(x)=ln(xsinx),Differentiatingusingchainruleweget,ddx(f(x))=1xsinxddx(xsinx).ddx(f(x))==1xsinx(xcosx+sinx)ddx(f(x))=(xcosx+sinx)xsinx=cotx+1x.

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