#### To determine

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

#### Answer

The first derivative of the function
f(x) is
1(3−x)2.

The second derivative of
f(x) is
2(3−x)3.

#### Explanation

**Given:**

The function
f(x)=13−x

**Derivative rules:**

(1) Constant Multiple Rule:
ddx[c⋅f(x)]=c⋅ddxf(x)

(2) Difference Rule:
ddx[f(x)−g(x)]=ddx(f(x))−ddx(g(x))

(3) Power Rule:
ddx(xn)=nxn−1

(4) Quotient Rule: If
f1(t) and
f2(t) are both differentiable, then

ddt[f1(t)f2(t)]=f2(x)ddt[f1(t)]−f1(t)ddt[f2(t)][f2(t)]2

**Calculation:**

The first derivative of
f(x) is
f′(x), which is obtained as follows,

f′(x)=ddx(f(x)) =ddx(13−x)

Apply the quotient rule (4),

f′(x)=(3−x)ddx(1)−(1)ddx(3−x)(3−x)2

Apply the constant multiple rule (1), power rule (3) and simplify the expressions.

f′(x)=(3−x)(0)−(1)[ddx(3)−ddx(x)](3−x)2=0−(1)[0−1](3−x)2=1(3−x)2

Therefore, the first derivative of the function
f(x) is
1(3−x)2.

The second derivative of
f(x) is
f″(x), which is obtained as follows,

f″(x)=ddx(f′(x))=ddx(1(3−x)2)

Apply the quotient rule (4),

f″(x)=((3−x))2ddx(1)−(1)ddx((3−x)2)(3−x)4

Apply the constant multiple rule (1), chain rule and power rule (3) and simplify the expressions.

f″(x)=(3−x)2[0]−(1)2(3−x)ddx((3−x))(3−x)4=0−2(3−x)(−1)(3−x)4=2(3−x)(3−x)4=2(3−x)3

Therefore, the second derivative of
f(x) is
2(3−x)3.