If f ( x ) = 4 x ( sin x + cos x ) , find f ' ( x ) .

Question

f'(x)=

Evaluate the derivative at x = 3. Enter an approximation, rounded to the nearest thousandth.
f'(3)=

Answer 1

use the product rule

f ' (x) = 4x(cosx - sinx) + 4(sinx + cosx)
f ' (3) = 12cos3 - 12sin3 + 4sin3 + 4cos3
= 16cos3 - 8sin3

Pretty hard to estimate, unless you make the very rough approximation of
π = 3
then f ' (3) = appr 16cosπ - 8sinπ
= roughly -16 + 0 = -16

actual answer: -16.968...

Interesting side note: On Feb. 6, 1897, Indiana's state representatives voted to declare 3.2 the legal value of pi to make things simpler.
The level of intellect of our politicians has not improved one digit since that time, when one looks at some of the performers from the southern states.

Answer 2

Just noticed that your question asked for an approximation, rounded to the nearest thousandth.
That is just plain absurd!