#### To determine

**To find:**

The linear density of the wire when x = 4 m

#### Answer

The linear density of the wire when x = 4 m is 4 kg/m

#### Explanation

**1. Concept:**

Use differentiation

**2. Formula:**

i. Product rule:

ddxfx*gx=fxddxgx+gxddx(fx)

ii. Radical rule:

ddxx=12x

iii. Sum rule:

ddxfx+gx=ddxfx+ddx(gx)

iv. Constant function rule:

ddxC=0

**3. Given:**

mx=x(1+x)

**4. Calculation:**

Linear density is the rate of change of mass with respect to the position at that point

Therefore,

Differentiate mass with respect to x

m'x=ddxx(1+x)

By using product rule,

m'x=xddx1+x+1+xddx(x)

By using sum rule,

m'x=xddx1+ddxx+1+xddx(x)

By using radical rule and constant function rule,

m'x=x0+12x+1+x

m'x=x2x+1+x

Now substitute x = 4

Therefore,

m'4=424+1+4=42*2+1+2=1+1+2=4 kg/m

**Conclusion:**

The linear density of the wire when x = 4 m is 4 kg/m