Derivatives in Differential Form

In Exercises 17–28, find dy.

y = (2√x)/(3(1 + √x))

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Step 1: Begin by identifying the function y = \(\frac{2\sqrt{x}\)}{3(1 + \(\sqrt{x}\))}. This is a rational function where both the numerator and the denominator involve square roots.

Step 2: Apply the quotient rule for derivatives, which states that if you have a function y = \(\frac{u(x)}{v(x)}\), then the derivative dy/dx is given by \(\frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}\). Here, u(x) = 2\(\sqrt{x}\) and v(x) = 3(1 + \(\sqrt{x}\)).

Step 3: Find the derivative of the numerator u(x) = 2\(\sqrt{x}\). The derivative u'(x) can be found using the chain rule. Recall that \(\sqrt{x}\) = x^{1/2}, so u'(x) = 2 \(\cdot\) \(\frac{1}{2}\)x^{-1/2} = x^{-1/2}.

Step 4: Find the derivative of the denominator v(x) = 3(1 + \(\sqrt{x}\)). The derivative v'(x) involves differentiating 1 + \(\sqrt{x}\). The derivative of \(\sqrt{x}\) is \(\frac{1}{2}\)x^{-1/2}, so v'(x) = 3 \(\cdot\) \(\frac{1}{2}\)x^{-1/2} = \(\frac{3}{2}\)x^{-1/2}.

Step 5: Substitute u(x), u'(x), v(x), and v'(x) into the quotient rule formula: dy/dx = \(\frac{x^{-1/2}\) \(\cdot\) 3(1 + \(\sqrt{x}\)) - 2\(\sqrt{x}\) \(\cdot\) \(\frac{3}{2}\)x^{-1/2}}{(3(1 + \(\sqrt{x}\)))^2}. Simplify the expression to find the derivative dy/dx.

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주요 개념

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Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that allows us to determine how a function behaves at any given point. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a specific point.

Differential Form

Differential form refers to the expression of derivatives in terms of differentials, typically denoted as dy and dx. In this context, dy represents the change in the function y as x changes, and is calculated using the derivative. This form is particularly useful for understanding how small changes in x affect y.

Quotient Rule

The quotient rule is a method for finding the derivative of a function that is the ratio of two other functions. If y = u/v, where u and v are functions of x, the derivative is given by dy/dx = (v(du/dx) - u(dv/dx)) / v². This rule is essential for differentiating functions like y = (2√x)/(3(1 + √x)), where both the numerator and denominator are functions of x.

Derivatives in Differential FormIn Exercises 17–28, find dy.y = (2√x)/(3(1 + √x))