#### To determine

**To find:** The first and second derivatives of the function.

#### Answer

The first derivative of the function G(r) is (12)r−12+(13)r−23.

The second derivative of G(r) is (−14)r−32+(−29)r−59.

#### Explanation

**Given:**

The function is G(r)=r+r3.

**Derivative rules:**

(1) Constant Multiple Rule: ddr[c⋅f(r)]=c⋅ddrf(r)

(2) Power Rule: ddr(rn)=nrn−1

(3) Sum Rule: ddr[f(r)+g(r)]=ddr(f(r))+ddr(g(r))

**Calculation:**

The first derivative of G(r) is G′(r), which is obtained as follows,

G′(r)=ddr(G(r)) =ddr(r+r3) =ddr(r12+r13)

Apply the sum rule (3),

G′(r)=ddr(r12)+ddr(r13)

Apply the power rule (2) and simplify the expression,

G′(r)=(12r12−1)+(13r13−1)=(12r12−22)+(13r13−33)=(12)r−12+(13)r−23

Therefore, the first derivative of the function G(r) is (12)r−12+(13)r−23 .

The second derivative of G(r) is G″(r), which is obtained as follows,

G″(r)=ddr(G′(r))=ddr((12)r−12+(13)r−23)

Apply the sum rule (3),

G″(r)=ddr(12r−12)+ddr(13r−23 )

Apply the constant multiple (1),

G″(r)=ddr(12r−12 )+ddr(13r−23 )=12ddr(r−12 )+13ddr(r−23 )

Apply the power rule (2) and simplify the terms,

G″(r)=12(−12r−12−1 )+13(−23r−23−1 )=12(−12r−12−22 )+13(−23r−23−33 )=12(−12r−32 )+13(−23r−53 )=(−14)r−32+(−29)r−59

Therefore, the second derivative of G(r) is (−14)r−32+(−29)r−59.