Problem 322 - Calculus Volume 1

Question

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

Step-by-Step Solution

Answer

\(\frac{dy}{dx} = e^{\sin x} \cos x\).

Step 1: Identify the Function

The given function is \[ y = e^{\sin x} \].Our task is to find \(\frac{dy}{dx}\) using implicit differentiation.

Step 2: Apply Natural Logarithm to Both Sides

Take the natural logarithm of both sides to simplify the differentiation process:\[ \ln y = \ln(e^{\sin x}) \].Using the property of logarithms, this simplifies to:\[ \ln y = \sin x \].

Step 3: Differentiate Both Sides with Respect to x

Differentiate the equation \(\ln y = \sin x\) with respect to \(x\):The derivative of \(\ln y\) with respect to \(x\) is \(\frac{1}{y} \frac{dy}{dx}\), due to the chain rule.The derivative of \(\sin x\) is \(\cos x\).Therefore, we get:\[ \frac{1}{y} \frac{dy}{dx} = \cos x \].

Step 4: Solve for dy/dx

Multiply both sides of the equation by \(y\) to solve for \(\frac{dy}{dx}\):\[ \frac{dy}{dx} = y \cos x \].Substitute back \(y = e^{\sin x}\) to get:\[ \frac{dy}{dx} = e^{\sin x} \cos x \].

Key Concepts

Implicit Differentiation

In differential calculus, implicit differentiation is a powerful technique used to find derivatives when a function is not expressed explicitly as \(y = f(x)\). Instead, the function may be given in a form where \(y\) and \(x\) are mixed, such as \(y^2 + x^2 = 1\).

To perform implicit differentiation:

• Differentiate both sides of the equation with respect to \(x\).
• Treat \(y\) as an implicit function of \(x\), meaning whenever you differentiate \(y\), multiply the derivative by \(\frac{dy}{dx}\) (or \(y'\)).
• After differentiating, rearrange the equation to solve for \(\frac{dy}{dx}\).

This technique allows us to differentiate complex equations that involve multiple variables and their interactions, which are not easily separable.

In the exercise, we applied implicit differentiation after taking the natural logarithm on both sides of the equation \(y = e^{\sin x}\), ultimately transforming it into a more manageable form for differentiation.

Chain Rule

The chain rule is an essential tool in calculus used to differentiate composite functions—a function made by applying one function to the result of another. For a function \(y = g(f(x))\), the chain rule states that the derivative \(\frac{dy}{dx}\) is the product of \(g'(f(x))\) and \(f'(x)\).

When you see \(y = e^{\sin x}\), note that this is a composite function: first, the sine function \(f(x) = \sin x\) is applied, followed by the exponential function \(g(u) = e^u\), where \(u = \sin x\). Using the chain rule:

• Differentiate \(e^{\sin x}\) to get \(e^{\sin x}\), since the derivative of \(e^u\) is \(e^u\).
• Then, differentiate \(\sin x\), which is \(\cos x\).
• Finally, multiply the results to apply the chain rule: \(e^{\sin x} \cdot \cos x\).

This approach simplifies finding the derivative of functions where actions occur in steps, making complex processes more manageable and straightforward.

Natural Logarithms

Natural logarithms, denoted \(\ln\), are logarithms with the base \(e\), where \(e\approx 2.718\). They are particularly useful in calculus for simplifying the process of differentiation, especially when dealing with exponential functions like \(e^{x}\).

A key property we use in calculus is that \(\ln(e^x) = x\). This logarithmic identity directly simplifies expressions, making differentiation easier. For instance:

• In the exercise, we apply \(\ln\) to transform \(e^{\sin x}\) into \(\sin x\), a simpler form for differentiation.
• It helps in reducing the exponentiation problem to a basic trigonometric expression.

Moreover, the derivative of \(\ln y\) in terms of \(x\) is \(\frac{1}{y}\frac{dy}{dx}\), showcasing how natural logs introduce the chain rule, facilitating the differentiation process further.

In conclusion, natural logarithms serve as a practical tool for handling functions involving \(e\), aiding in their differentiation by breaking down complex expressions into simpler, manageable forms.