Proof Let u be a differentiable function of x . Use the fact that |u|=√(u^2) to prove that (d)/(

step 1 In this problem we're asked to prove that the derivative of u, which is the function of x, is eq

Proof Let u be a differentiable function of x . Use the fact...

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Key Concepts

Absolute Value Function

The absolute value function, denoted as |u|, gives the non-negative magnitude of a real number. Expressing |u| as ?(u²) is a key insight that allows one to employ differentiation techniques applicable to square roots, thus facilitating the exploration of its derivative when u is nonzero.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that if a function is composed of two or more nested functions, the derivative of the overall function is the product of the derivative of the outer function, evaluated at the inner function, and the derivative of the inner function.

Differentiation of the Square Root Function

When differentiating a function of the form ?(v), the derivative is (1/(2?(v))) multiplied by the derivative of v. This result is utilized by recognizing |u| as ?(u²) and then applying the chain rule to differentiate the square root of u².

Differentiability Conditions

To correctly apply these differentiation techniques, it is essential that u is differentiable and that u ? 0. The condition u ? 0 is critical because the derivative formula involves division by |u|, which would be undefined at the point u = 0.

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