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Differentiate f(x)=3x+x2in three ways. When you have completed all three parts, show that your three answers are the same:

(a) with the chain rule

(b) with the product rule but not the chain rule

(c) without the chain or product rules.

Short Answer

Expert verified

The derivative of f(x)using each rule is:localid="1648368040492" f'(x)=18x+9x+1

Step by step solution

01

Step 1. Given information:

The function is:

f(x)=3x+x2
02

Part (a). Step 1. To find derivative using chain rule:

f(x)=3x+x2

Let u=3x+x

then f(u)=u2

Derivative of uwith respect to x:

dudx=ddx(3x+x)=3+12x

Derivative of fwith respect to u:

dfdu=dduu2=2u=2(3x+x)

Derivative of fwith respect to x:

localid="1648368008391" dfdx=dfdu×dudx=23x+x3+12x=29x+32x+3x+12=29x+92x+12=18x+9x+1

03

Part (b). Step 1. To find derivative using product rule:

f(x)=3x+x2=3x+x.3x+x

U=3x+xV=3x+x

localid="1648367311339" dUdx=ddx(3x+x)=3+12x=dVdx

Now derivative of a function using the product rule is:

localid="1648367986645" dfdx=UdVdx+VdUdx=3x+x3+12x+3x+x3+12x=23x+x3+12x=29x+32x+3x+12=29x+92x+12=18x+9x+1

04

Part (c). Step 1. To find derivative without using chain rule or product rule:

f(x)=3x+x2=3x2+x2+23xx=9x2+x+6x32

Now differentiate this with respect to x:

dfdx=ddx9x2+x+6x32=9.2x+1+6.32x12=18x+1+9x

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