Expert: Scotto - 5/11/2009

Question
For this:
4x(sqrt(4x^2+2))

How do I get the derivative?? I used the product law, then the chain rule. I can't figure out how to get it down so that you can find the critical numbers.

(A) Find all critical values of f.
Critical value(s) =

(B) Use interval notation to indicate where f(x) is increasing.

(C) Use interval notation to indicate where f(x) is decreasing.  

Answer
It looks like the product of two functions, call them f(x) and g(x).

The definition of the two functions is f(x) = 4x and g(x) = √(4x²+2).
The derivative is f(x)g'(x) + g(x)f'(x).

Since f(x) is 4x, it can be seen that f'(x) = 4.

The function g(x) can be thought of as g(t(x)) where g(t) = √t and
t(x) = 4x² + 2.  From here, we know that g'(t) = 1/(2√t).
We can also see that t'(x) is 8x.

What you have at the start is f(x)g(t(x)).
The derivative is f'(x)g(t(x)) + f(x)g'(t)t'(x).

This is where f'(x) is more easily understood as df/dx .

If we have the function y(x) = f(x)g(t(x)), then
dy/dx = (df/dx)g(x) + f(x)(dg/dt)(dt/dx).
In this way, the 'dt' can be seen to 'cancel' in the second term of the fraction, so you end up with dg/dx.

All told, the derivative is 4√(4x²+2) + 4x*(1/(2√(4x²+2)))8x.
ON the second term, the 4x at the start times the 8x at the end is 32x².  Since this is divided by 2, the top has a 16.  Once this has been done, some factor a 4 out of both terms and put the 4 out front with parenthesis around both terms.  Others don't like the √ being in the denominator of the second term, so the second term is converted by multiplying the top and bottom by √(4x²+2).

(A) when f'(t) is 0
(B) it is increasing when f'(t)>0
(C) it is decreasing when f'(t)<0.

Find f'(t) first.  Set it to 0 and solve.
The intervals will be the point below the first point where the derivative is 0, between each of the points, and above the last point where it is zero.  Check a value in each of the intervals in f'(x) to determine whether the function is increasing or decreasing.