Problem 322 - Calculus Volume 1

Question

For the following exercises, compute \(d y / d x\) by differentiating \(\ln y\). $$y=e^{\sin x}$$

Step-by-Step Solution

Answer

The derivative \(dy/dx\) is \(e^{\sin x} \cdot \cos x\).

Step 1: Take the Natural Logarithm

Start by taking the natural logarithm of both sides of the given equation. This will help us to differentiate using the properties of logarithms. The equation becomes: \[\ln y = \ln(e^{\sin x})\]

Step 2: Simplify the Logarithmic Equation

Simplify the right-hand side of the equation using the logarithmic identity \(\ln(e^a) = a\). Therefore, the equation becomes: \[\ln y = \sin x\]

Step 3: Differentiate Both Sides with Respect to x

Now, differentiate both sides of the equation with respect to \(x\). For the left-hand side, use the chain rule: \[\frac{d}{dx}(\ln y) = \frac{1}{y} \cdot \frac{dy}{dx}\]For the right-hand side, the derivative of \(\sin x\) is \(\cos x\), so we have: \[\frac{1}{y} \cdot \frac{dy}{dx} = \cos x\]

Step 4: Solve for dy/dx

To find \(\frac{dy}{dx}\), multiply both sides by \(y\): \[\frac{dy}{dx} = y \cdot \cos x\]Since \(y = e^{\sin x}\), substitute back for \(y\): \[\frac{dy}{dx} = e^{\sin x} \cdot \cos x\]

Key Concepts

Chain Rule

The Chain Rule is a fundamental concept in calculus used to find the derivative of a composite function. When you have a function composed of other functions, the chain rule allows you to differentiate it step by step.

Let's imagine you have a function \( y = f(g(x)) \). Here, \( f \) is applied to \( g(x) \), which means that first \( g(x) \) is evaluated, and then \( f \) is applied to its result. To find the derivative of \( y \) with respect to \( x \), use:

• Differentiate \( f \) with respect to \( g(x) \), which is \( f'(g(x)) \).
• Multiply it by the derivative of \( g(x) \) with respect to \( x \), which is \( g'(x) \).

Mathematically, this is expressed as:\[\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\]In the solution above, when differentiating \( \ln y \) in terms of \( x \), the chain rule is applied: \( \frac{d}{dx}(\ln y) = \frac{1}{y} \cdot \frac{dy}{dx} \). This ensures every part of the function is accounted for.

Natural Logarithm

The Natural Logarithm, denoted as \( \ln \), is a logarithm with base \( e \), where \( e \) is an irrational constant approximately equal to 2.718. The natural logarithm has unique properties that make differentiation and integration of expressions simpler.

Some properties of the natural logarithm that are useful in calculus include:

• \( \ln(e^a) = a \), which is used to simplify expressions involving exponentials.
• \( \ln(xy) = \ln x + \ln y \), which helps break down multiplication expressions.
• \( \ln\left(\frac{x}{y}\right) = \ln x - \ln y \), simplifying division.

In the exercise, taking the natural log of both sides \( \ln y = \ln(e^{\sin x}) \) lets us leverage the first property \( \ln(e^a) = a \), simplifying our task to just differentiating \( \sin x \). This step significantly reduces complexity in finding derivatives.

Exponential Functions

Exponential Functions are functions of the form \( y = a^{x} \) where \( a \) is a constant. The base \( e \) is special because it simplifies derivatives and integrals. When you have \( y = e^u \) where \( u \) is a function of \( x \), the differentiation process is straightforward.

The derivative of \( e^x \) with respect to \( x \) is \( e^x \), which is unique and convenient for calculus. This property extends to any function of the form \( e^{f(x)} \) where the derivative becomes:

• \( \frac{dy}{dx} = e^{f(x)} \cdot f'(x) \), using the chain rule.

In this exercise, \( y = e^{\sin x} \), making \( e \) the base and \( \sin x \) the exponent. Differentiating means applying both the exponential and the chain rule properties, resulting in \( e^{\sin x} \cdot \cos x \). The natural interaction between the exponential and trigonometric functions showcases why \( e \) is such a fundamental constant in calculus.