#### To determine

**To find:** The differentiation of the function
R(a)=(3a+1)2.

#### Answer

The differentiation of the function
R(a)=(3a+1)2 is
18a+6_.

#### Explanation

**Given:**

The function,
R(a)=(3a+1)2.

**Formula used:**

**Derivative of a Constant Function:**

If *c* is a constant function, Then
dda(c)=0
(1)

**The Constant Multiple Rule:**

If *c* is a constant and
f(a) is a differentiable function, then the constant multiple rule is,

dda[c⋅f(a)]=c⋅ddaf(a) (2)

**The Power Rule:**

If n is any real number, then the power rule is,

dda(an)=nan−1 (3)

**The Sum Rule:**

If
f(a) and
g(a) are both differentiable functions, then the sum rule is,

dda[f(a)+g(a)]=dda(f(a))+dda(g(a)) (4)

**The Square of Binomial:**

(m+n)2=m2+n2+2mn (5)

**Calculation:**

The derivative of
R(a) is
R′(a) as follows.

R′(a)=dda[R(a)] =dda[(3a+1)2]

Expand the expression using equation (5).

R′(a)=dda(9a2+1+(2×3a×1))=dda(9a2+1+6a)

Apply the sum rule as shown in equation (4).

R′(a)=dda(9a2)+dda(1)+dda(6a)

Apply the constant multiple rule as shown in equation in (2).

R′(a)=9dda(a2)+dda(1)+6dda(a)

Apply the power rule as shown in equation in (3).

R′(a)=9(2a2−1)+dda(1)+6(1a1−1)

Apply the rule of derivative of constant function as shown in equation (1).

R′(a)=9(2a)+0+6(1a0)=18a+6

Therefore, the differentiation of the function
R(a)=(3a+1)2 is
18a+6.