Evaluate the given expression.


d
dx
[2(x2 + 3|x|)]
1

I am guessing you want the derivative of the piecewise-continuous function

f(x) = 2(x^2 + 3|x|) ,

but what does the "1" represent?

If you want the derivative at x=1, you can remove the | | signs

For |x|, we have 2 situations.

|x| = x, for x >= 0
|x| = -x, for x < 0

So:
¤ for x>=0:
d/dx (2(x^2 + 3x)) = ?
¤ for x<0:
d/dx (2(x^2 - 3x)) = ?

To evaluate the given expression, we need to find the derivative of the function 2(x^2 + 3|x|) with respect to x and then substitute the value 1 into the resulting derivative function.

Step 1: Find the derivative of the expression.
The given expression can be rewritten as 2(x^2 + 3|x|) = 2x^2 + 6|x|.

To differentiate 2x^2, we apply the power rule, which states that the derivative of x^n (where n is a constant) is nx^(n-1). In this case, n = 2, so we get: d(2x^2)/dx = 4x.

To differentiate 6|x|, we use the chain rule, which states that if we have a composite function f(g(x)), the derivative is given by f'(g(x)) * g'(x). In this case, f(g(x)) = 6|x| and g(x) = |x|. So, f'(g(x)) = 6 and g'(x) is the derivative of |x|.

The derivative of |x| is not defined at x = 0, but we can find its derivative for x < 0 and x > 0 separately.

For x < 0, |x| = -x, so d(-x)/dx = -1.

For x > 0, |x| = x, so d(x)/dx = 1.

Therefore, the derivative of 6|x| is -6 for x < 0 and 6 for x > 0.

Putting it all together, the derivative of 2(x^2 + 3|x|) is 4x + 6 when x > 0 and 4x - 6 when x < 0.

Step 2: Evaluate the derivative at x = 1.
Substitute x = 1 in the derivative function to get:
(4 * 1) + 6 = 4 + 6 = 10.

Therefore, the value of the given expression when x = 1 is 10.