#### To determine

**To find:** The derivative of the function G(q)=(1+q−1)2.

#### Answer

The derivative of the function G(q)=(1+q−1)2 is −2q−3(1+q)_.

#### Explanation

**Given:**

The function, G(q)=(1+q−1)2.

**Formula used:**

**Derivative of a Constant Function**

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

**The Constant Multiple Rule**

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

ddq[c⋅f(q)]=c⋅ddqf(q) (2)

**The Power Rule:**

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

ddq(qn)=nqn−1 (3)

**The Sum Rule:**

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

ddq[f(q)+g(q)]=ddq(f(q))+ddq(g(q)) (4)

**The Square of Binomial:**

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

**Calculation:**

The derivative of G(q) is G′(q) as follows:

G′(q)=ddq(G(q)) =ddq[(1+q−1)2]=ddq[(1+1q)2]

Expand the expression by using equation (5).

G′(q)=ddq(1+1q2+(2⋅1⋅1q)) =ddq(1+q−2+2q−1)

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

G′(q)=ddq(1)+ddq(q−2)+ddq(2q−1)

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

G′(q)=ddq(1)+ddq(q−2)+2ddq(q−1)

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

G′(q) =ddq(1)+(−2q−2−1)+2(−1q−1−1)

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

G′(q)=0+(−2q−3)+2(−1q−2)=(−2q−3)−2(q−2)=−2q−3−2q−3q=−2q−3(1+q)

Therefore, the derivative of the function G(q)=(1+q−1)2 is −2q−3(1+q)_.