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Differentiation review: Differentiate each of the functions that follow. Simplify your answers as much as possible.

a. f(x)=xexex

b. f(x)=x2lnx2x24

c. f(x)=e-x(x2+1)2xe-x2e-x

d.f(x)=x332(xexex)+e2x2

e. f(x)=xcotx+ln|sinx|

f. f(x)=x3sinx+3x2cosx6xsinx6cosx

Short Answer

Expert verified

Part (a)f'(x)=xex

Part (b)f'x=xlnx

Part (c)f'(x)=x2e-x+e-x

Part (d)f'(x)=x22xex+e2x

Part (e)f'(x)=xcsc2x

Part (f)f'(x)=x3cosx

Step by step solution

01

Part (a) Step 1. Finding the derivative

Given function, f(x)=xexex

The derivative, f'(x)=xex+ex1-ex

f'(x)=xex

02

Part (b) Step 1. Finding the derivative

Given function,f(x)=x2lnx2x24

The derivative, f'(x)=x21x+lnx2x22x4

f'x=x2+xlnx-x2f'x=xlnx

03

Part (c) Step 1. Finding the derivative

Given function, f(x)=e-x(x2+1)2xe-x2e-x

f(x)=e-xx2-e-x2xe-x2e-x

The derivative, role="math" localid="1653980597687" f'(x)=e-x2x+x2(-e-x)--e-x2x-e-x+e-x(1)2-e-x

role="math" localid="1653982097270" f'(x)=2xe-x+x2e-x+e-x+2xe-x-2e-x+2e-xf'(x)=x2e-x+e-x

04

Part (d) Step 1. Finding the derivative

Given function,f(x)=x332(xexex)+e2x2

The derivative, f'(x)=3x232xex+ex(1)ex+2e2x2

f'(x)=x22xex+exex+e2xf'(x)=x22xex+e2x

05

Part (e) Step 1. Finding the derivative

Given function,f(x)=xcotx+ln|sinx|

The derivative, f'(x)=x(-csc2x)-cotx(1)+1sinx×cosx

f'(x)=xcsc2x-cotx+cotxf'(x)=xcsc2x

06

Part (f) Step 1. Finding the derivative

Given function,f(x)=x3sinx+3x2cosx6xsinx6cosx

The derivative, f'(x)=x3cosx+sinx(3x2)+3x2(-sinx)+cosx(2x)6xcosx+sinx(1)6(-sinx)

f'(x)=x3cosx+3x2sinx-3x2sinx+6xcosx6xcosx-6sinx+6sinxf'(x)=x3cosx

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