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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)=3x2lnxtanx.

Short Answer

Expert verified

The derivative of the given function is:3xtanx(1+2lnx)-3x2lnxsec2xtan2x.

Step by step solution

01

Step 1. Given Information.

f(x)=3x2lnxtanx.

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)=3x2lnxtanx,Differentiatingusingproductandquotientruleweget,ddx(f(x))=tanxddx(3x2lnx)-(3x2lnx)ddx(tanx)tan2x=tanx(3x21x+6xlnx)-(3x2lnx)(sec2x)tan2x=tanx(3x+6xlnx)-(3x2lnx)sec2xtan2x=3xtanx(1+2lnx)-3x2lnxsec2xtan2x.

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