91Ó°ÊÓ

The quotient rule is a fundamental tool in calculus used to differentiate functions that are ratios of two differentiable functions. When working with the quotient of two functions, say \( u(x) = e^x - e^{-x} \) and \( v(x) = e^x + e^{-x} \), you need to apply the quotient rule to find the derivative. The quotient rule is defined as:

• \( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} \)

To use it, identify which part of your function is \( u \) (the numerator) and \( v \) (the denominator). Then, compute the derivatives of \( u \) and \( v \), and substitute them back into the formula. This step-by-step approach ensures that you accurately compute the derivative of complex fractions.

Exponential functions are a special class of functions where a constant base is raised to a variable exponent, generally represented as \( e^x \). Here, both the numerator and the denominator of the function are composed of exponential functions. When differentiating exponential functions, remember the basic derivative rule:

• The derivative of \( e^x \) is \( e^x \).
• For negative exponents like \( e^{-x} \), use the chain rule to get \(-e^{-x}\).'

In our problem, for \( u = e^x - e^{-x} \), its derivative is the addition of the individual derivatives: \( \frac{du}{dx} = e^x + e^{-x} \). Meanwhile, for \( v = e^x + e^{-x} \), its derivative is \( \frac{dv}{dx} = e^x - e^{-x} \). These exponential identities and rules are crucial when substituting derivatives back into the quotient rule formula.

Simplification is often a necessary part of finding the derivative, aimed at making the result more understandable or elegant. After applying the quotient rule, you'll typically end up with a complex expression. In this case, our initial expression for the derivative was:

• \( \frac{(e^x + e^{-x})(e^x + e^{-x}) - (e^x - e^{-x})(e^x - e^{-x})}{(e^x + e^{-x})^2} \)

The simplification process often involves expanding and re-factoring the expression, as seen when the squared terms in the numerator were individually expanded and subtracted. By doing so, all terms cancel out except the constants, leading to \( 4 \). The final, simplified expression for the derivative appeared much cleaner:

• \( \frac{4}{(e^x + e^{-x})^2} \)

Such simplification allows for a simpler expression which is easier to interpret and often easier to work with in further calculations.