Evaluate the given expression:
d/dx[5(x^2 + 3|x|)]
Any help with this is appreciated!
|x| is a tricky one, since it changes at x=0. Consider it a |x| = √x^2. Then, using the chain rule,
d/dx √x^2 = 2x/(2√(x^2)) = x/|x|
Note that this correctly captures the change of slope at x=0.
d/dx[5(x^2 + 3|x|)]
= 5(2x + 3x/|x|)
Also, the function is not differentiable at x=0.
Thank you!
To evaluate the given expression, we will use the derivative rules. Specifically, we will use the sum rule, product rule, and the chain rule if necessary.
Let's break down the given expression step by step:
5(x^2 + 3|x|)
The expression contains two terms: x^2 and 3|x|. To find the derivative of each term, we apply the power rule for x^2 and the derivative of absolute value for 3|x|.
1. Term: x^2
The derivative of x^2 with respect to x is calculated as follows:
d/dx(x^2) = 2x
2. Term: 3|x|
To find the derivative of 3|x|, we consider two cases: when x > 0 and when x < 0.
Case 1: x > 0
For this case, the absolute value function simplifies to x.
So, the derivative of 3|x| with respect to x is:
d/dx(3x) = 3
Case 2: x < 0
For this case, the absolute value function simplifies to -x.
So, the derivative of 3|x| with respect to x is:
d/dx(-3x) = -3
Now that we have found the derivatives of each term, we can put it all together using the sum rule.
The sum rule states that the derivative of the sum of two functions is the sum of their derivatives.
Therefore, applying the sum rule:
d/dx[5(x^2 + 3|x|)] = d/dx(5x^2) + d/dx(15|x|) = 10x + 15|x|
And that is the final result of the given expression.