Use Version 1 of the Chain Rule to calculate (d y)/(d x)....

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Key Concepts

Exponential Function Differentiation

Differentiating an exponential function, like e^(u), involves recognizing that the derivative of e^(u) with respect to u is e^(u) itself. When the exponent u is a function of x, the chain rule necessitates multiplying this by the derivative of u with respect to x.

Square Root Differentiation

The square root function, expressed as sqrt(x) or x^(1/2), is differentiated using the power rule. The derivative of x^(1/2) is (1/2)x^(-1/2), which simplifies to 1/(2?x). This is essential when the square root appears as an inner function within a composite function.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y can be expressed as a composition of two functions, such as y = f(g(x)), then its derivative is given by f'(g(x)) multiplied by g'(x). This method allows us to break down complex differentiations into simpler parts.

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