#### To determine

**To find:** The differentiation of the function
f(x)=(5x2−2)(x3+3x).

#### Answer

The differentiation of the function
f(x) is
f′(x)=25x4+39x2−6 .

#### Explanation

**Definitions used:**

**Derivative rule:**

(1) Product Rule**:** If
f(x) and
g(x) are both differentiable, then

ddx[f(x)g(x)]=f(x)ddx[g(x)]+g(x)ddx[f(x)]

(2) Constant multiple rule:
ddx(cf)=cddx(f)

(3) Power rule:
ddx(xn)=nxn−1

(4) Difference rule:
ddx(f−g)=ddx(f)−ddx(g)

**Calculation:**

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

f′(x)=ddx(f(x))=ddx[(5x2−2)(x3+3x)]

Substitute
(5x2−2) for
f(x) and
(x3+3x) for
g(x) in the product rule (1),

f′(x)=(5x2−2)ddx(x3+3x)+(x3+3x)ddx(5x2−2)

Apply the constant multiple rule (2) and the power rule (3),

f′(x)=(5x2−2)[ddx(x3)+3ddx(x)]+(x3+3x)[5ddx(x2)−0]=(5x2−2)(3x3−1+3)+(x3+3x)[5(2)x2−1]=(5x2−2)(3x2+3)+(x3+3x)[10x]=15x4+15x2−6x2−6+10x4+30x2

Simplify further as
15x4+15x2−6x2−6+10x4+30x2=25x4+39x2−6

Therefore, the differentiation of the function
f(x) is
f′(x)=25x4+39x2−6.