#### To determine

**To find:** The derivative of the function y=x+x+x.

#### Answer

The derivative of y=x+x+x is 1+1+12x2x+x2x+x+x_.

#### Explanation

**Given:**

The function is y=x+x+x.

**Derivative rule used:**

**Chain Rule:**

If *h* is differentiable at *x* and *g* is differentiable at h(x), then the composite function F=g∘h defined by F(x)=g(h(x)) is differentiable at *x* and F′ is given by the product

F′(x)=g′(h(x))⋅h′(x) (1)

**Power Rule:**

If *n* is positive integer, then ddx(xn)=nxn−1 (2)

**Calculation:**

Obtain the derivative y.

y′=ddx(y)=ddx(x+x+x)

Let h(x)=x+x+x and g(u)=u where u=h(x)

Apply the chain rule as shown in equation (1),

y′(x)=g′(h(x))⋅h′(x) (3)

The derivative g′(h(x)) is computed as follows,

g′(h(x))=g′(u)=ddu(g(u))=ddu(u)=ddu(u)12

Apply the power rule as shown in equation (2),

g′(h(x))=12(u)12−1=12(u)1−22=12(u)−12=12u

Substitute u=x+x+x in the above equation,

g′(h(x))=12x+x+x.

Thus, the derivative of g′(h(x))=12x+x+x.

The derivative of h(x) is computed as follows,

h′(x)=ddx(x+x+x)=ddx(x)+ddx(x+x)

h′(x)=1+ddx(x+x) (4)

Obtain the derivative of ddx(x+x)

Let k(x)=x+x and s(u)=u where u=k(x)

Apply the chain rule as shown in equation (1),

ddx(x+x)=s′(k(x))⋅k′(x) (5)

The derivative of s′(k(x)) computed as follows,

s′(k(x)) =s′(u)=ddu(s(u))=ddu(u)=ddu(u)12

Apply the power rule as shown in equation (2),

s′(k(x))=12(u)12−1=12(u)1−22=12(u)−12=12u

Substitute u=x+x in the above equation,

s′(k(x))=12x+x

Thus, the derivative of s′(k(x))=12x+x.

The derivative of k(x) is computed as follows,

k′(x)=ddx(x+x)=ddx(x)+ddx(x)

Apply the power rule as shown in equation (2),

k′(x)=(1x1−1)+(12x12−1)=1+12x1−22=1+12x−12=1+12x

Thus, the derivative of k′(x)=1+12x.

Substitute 12x+x for s′(k(x)) and 1+12x for k′(x) in equation (5),

ddx(x+x)=12x+x(1+12x)=1+12x2x+x

Substitute 1+12x2x+x for ddx(x+x) in equation (4),

h′(x)=1+ddx(x+x)=1+1+12x2x+x

Substitute 12x+x+x for g′(h(x)) and 1+1+12x2x+x for h′(x) in equation (3),

g′(h(x))⋅h′(x)=12x+x+x(1+1+12x2x+x)=1+1+12x2x+x2x+x+x

Therefore, The derivative of y=x+x+x is 1+1+12x2x+x2x+x+x_.