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

f'(x)=

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

Just noticed that your question asked for an approximation, rounded to the nearest thousandth.

That is just plain absurd!

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.

To find the derivative of f(x) = 4x(sin x + cos x), we can use the product rule and the chain rule. The product rule states that if we have two functions u(x) and v(x), the derivative of their product is given by:

(d/dx)(u(x) * v(x)) = u'(x) * v(x) + u(x) * v'(x)

In this case, u(x) = 4x and v(x) = (sin x + cos x). We can find the derivatives of u(x) and v(x) as follows:

u'(x) = 4 * 1 = 4

To find v'(x), we need to use the chain rule. The chain rule states that if we have a composite function g(f(x)), the derivative is given by:

(d/dx)(g(f(x))) = g'(f(x)) * f'(x)

In this case, g(x) = sin x + cos x. The derivative of g(x) = sin x + cos x with respect to x is:

g'(x) = (d/dx)(sin x + cos x) = cos x - sin x

Now, we can find v'(x) by substituting g'(x) = cos x - sin x and f'(x) = 4 into the chain rule:

v'(x) = (cos x - sin x) * 4

Now, we can substitute the values of u'(x) = 4 and v'(x) = (cos x - sin x) * 4 into the product rule:

f'(x) = u'(x) * v(x) + u(x) * v'(x)
= 4 * (sin x + cos x) + 4x * (cos x - sin x)

Simplifying, we get:

f'(x) = 4sin x + 4cos x + 4xcos x - 4xsin x

To evaluate the derivative at x = 3, we substitute x = 3 into f'(x) and round the result to the nearest thousandth:

f'(3) = 4sin(3) + 4cos(3) + 4(3)cos(3) - 4(3)sin(3)

Evaluating this expression, we get the approximation for f'(3).