To determine

To find: The differentiation of the function B(u)=(u3+1)(2u2−4u−1).

Answer

The differentiation of the function B(u) is B′(u)=10u4−16u3−3u2+4u−4.

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 B(u) is B′(u), which is obtained as follows,

B′(u)=ddu(B(u))=ddu[(u3+1)(2u2−4u−1)]

Substitute (u3+1) for f(x) and (2u2−4u−1) for g(x) in the product rule (1),

B′(u)=(u3+1)ddu(2u2−4u−1)+(2u2−4u−1)ddu(u3+1)

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

B′(u)=(u3+1)[2ddu(u2)−4ddu(u)−ddu(1)]+(2u2−4u−1)[ddu(u3)+ddu(1)]=(u3+1)[2(2)u2−1−4(1)−0]+(2u2−4u−1)[3u3−1+0]=(u3+1)[4u−4]+(2u2−4u−1)[3u2]=4u4+4u−4u3−4+6u4−12u3−3u2

Simplify further as 4u4+4u−4u3−4+6u4−12u3−3u2=10u4−16u3−3u2+4u−4

Therefore, the differentiation of the function B(u) is B′(u)=10u4−16u3−3u2+4u−4.