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Suppose u(x)=3x2+1and f(u)=u2+3u51-u. Use the chain rule to find role="math" localid="1648356625815" ddx(f(u(x))) without first finding the formula for f(u(x)).

Short Answer

Expert verified

The derivative of the function is:3x45x23x2+1+143x2+1-36x4-72x2-101-3x2+12

Step by step solution

01

Step 1. Given information:

The functions are:

u(x)=3x2+1

f=u2+3u51-u

The composite function is:f(u(x))

02

Step 2. Find derivative of u(x) using chain rule.

Since, u(x)=3x2+1u(x)=(3x2+1)12

Let v=3x2+1

Then u=v12

Derivative of u(v)with respect to v:

dudv=12(v)12-1=12(v)-12=12v=123x2+1

Derivative of v(x)with respect to x:

dvdx=ddx(3x2+1)=3.2x+0=6x

dudx=dudv×dvdx=123x2+1×6x=3x3x2+1

03

Step 3. Find the derivative of f(u):

f(u)=u2+3u51-u

Differentiatedfdu=(1-u)ddu(u2+3u5)-(u2+3u5)ddu(1-u)(1-u)2=(1-u)(2u+15u4)-(u2+3u5)(-1)(1-u)2=2u+15u4-2u2-15u5+u2+3u5(1-u)2=2u+15u4-u2-12u5(1-u)2

04

Step 4. Find derivative of f(u(x)):

By using the chain rule, the derivative of the function is:

ddx(f(u(x)))=dfdu×dudx

Substitute the values from step 3 and 4.

ddxf(u(x))=2u+15u4-u2-12u5(1-u)2×3x3x2+1

Substitute value of u(x)=3x2+1and simplify:

ddx(f(u(x)))=23x2+1+153x2+14-3x2+12-123x2+151-3x2+12×3x3x2+1=3x2+12+153x2+13-3x2+1-123x2+141-3x2+12×3x3x2+1=3x2+153x2+13-3x2+1-123x2+141-3x2+12=3x2+153x2+13x2+1-3x2+1-123x2+121-3x2+12=3x2+45x23x2+1+153x2+1-3x2+1-123x4+1+6x21-3x2+12=3x2+45x23x2+1+143x2+1-36x4-12-72x21-3x2+12=3x45x23x2+1+143x2+1-36x4-72x2-101-3x2+12

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