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.