To determine
To find: The value of
f′(4).
Answer
The value is
f′(4)=16.
Explanation
Given:
The function
f(x)=xg(x).
The values
g(4)=8 and g′(4)=7.
Derivative rule:
(1) Product Rule:
ddx[f1(x)f2(x)]=f1(x)ddx[f2(x)]+f2(x)ddx[f1(x)]
(2) Power Rule:
ddx(xn)=nxn−1
Calculation:
Obtain the value of
f′(4).
f′(x)=ddx(xg(x))
Apply the product rule (1) and power rule (2),
f′(x)=x⋅ddx(g(x))+g(x)⋅ddx(x)=x12⋅ddx(g(x))+g(x)⋅ddx(x12)=x12g′(x)+g(x)(12)x12−1=x12g′(x)+g(x)(12)x−12
Substitute 4 for x and simplify the terms,
f′(4)=(4)12g′(4)+g(4)(12)(4)−12=(22)12g′(4)+g(4)(12)(22)−12
Substitute the values
g(4)=8 and g′(4)=7,
f′(4)==2(7)+(8)(12)(2−1)=14+42=14+2=16
Therefore, the value is
f′(4)=16.