Chain Rule Of Differentiation For Absolute Value: Derivatives Of U With Respect To X

d/dx[|u|]

(u/|u|)u’,u≠0

The function |u| represents the absolute value of u, which is defined as:

|u| = {u, if u >= 0
{-u, if u < 0 To differentiate |u| with respect to x, we need to apply the chain rule of differentiation. Let's assume that u is a function of x, i.e., u = f(x). Then, d/dx[|u|] = d/dx[u], if u >= 0
d/dx[-u], if u < 0 To determine which case applies, we need to look at the sign of u. If u is positive or zero, then |u| = u, and we use the first case. If u is negative, then |u| = -u, and we use the second case. Therefore, we can write: d/dx[|u|] = d/dx[u], if u >= 0
-d/dx[u], if u < 0 Now, we can use the chain rule to differentiate u with respect to x. Let's write the full expression: d/dx[|u|] = d/dx[u], if u >= 0
-d/dx[u], if u < 0 = d/dx[f(x)], if f(x) >= 0
-d/dx[f(x)], if f(x) < 0 Finally, we can simplify the expression by writing the sign of f(x) explicitly: d/dx[|u|] = d/dx[f(x)], if f(x) >= 0
-d/dx[f(x)], if f(x) < 0 In other words, the derivative of |u| with respect to x is equal to the derivative of u (or f(x)) if u (or f(x)) is positive or zero, and the negative of the derivative of u (or f(x)) if u (or f(x)) is negative. Note that if u is not a differentiable function (e.g., u = |x|), then d/dx[|u|] may not exist at certain points.

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