   Chapter 3.1, Problem 46E

Chapter
Section
Textbook Problem

Find the first and second derivatives of the function. G ( r ) = r   +   r 3

To determine

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

Explanation

Given:

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

Derivative rules:

(1) Constant Multiple Rule: ddr[cf(r)]=cddrf(r)

(2) Power Rule: ddr(rn)=nrn1

(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)=(12r121)+(13r131)=(12r1222)+(13r1333)=(12)r12+(13)r23

Therefore, the first derivative of the function G(r) is (12)r12+(13)r23

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