91Ó°ÊÓ

Understanding how to differentiate exponential functions is crucial in calculus. Exponential functions are of the form \( e^y \), where \( e \) is Euler's number, approximately equal to 2.71828. When differentiating \( e^y \) with respect to \( x \), we utilize implicit differentiation. This involves applying the chain rule, as \( y \) is a function of \( x \).
To perform implicit differentiation, take the derivative of \( e^y \). The derivative of \( e^y \) is \( e^y \) multiplied by the derivative of \( y \) with respect to \( x \), symbolized by \( \frac{dy}{dx} \). Therefore, the expression becomes \( \frac{d}{dx}e^y = e^y \cdot \frac{dy}{dx} \).
This approach helps differentiate functions where variables are intertwined, making it a robust method for many types of equations.

The slope of a curve at any given point is essentially the derivative at that point. It represents the rate of change or the steepness of the curve. For implicit functions like \( x = e^y \), the slope is found after differentiating and solving for \( \frac{dy}{dx} \).
Sometimes, we need to evaluate this derivative at a particular point. For instance, in the original exercise, at the point \((2, \ln 2)\), the expression \( \frac{dy}{dx} = \frac{1}{e^y} \) is evaluated by substituting \( y = \ln 2 \).

• Substitute \( y = \ln 2 \) into the derivative expression \( \frac{dy}{dx} \).
• Recognize that \( e^{\ln 2} = 2 \).
• This simplifies to \( \frac{1}{2} \), which is the slope at the given point.

Understanding the slope provides insights into how the function behaves at any small interval around this point.

Rewriting equations involves expressing one variable explicitly in terms of another. This is especially helpful when dealing with implicit functions where variables are mixed. Take for instance the equation \( x = e^y \). Here, \( x \) is expressed in terms of \( y \).
To make the differentiation process easier, rewrite such that you understand the dependent relationship. You know that \( e^y \) must somehow represent a transformation of \( x \).

• Isolate \( y \) if needed by using logarithms, as with \( y = \ln x \) from \( x = e^y \).
• Manipulating equations this way helps prepare for differentiation.
• Clear expressions simplify finding derivatives and understanding the slope as it links back to a known function.

Rewriting equations is a fundamental skill that makes it easier to navigate challenges in calculus and find solutions efficiently.