Problem 26 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(D_{x} y\). $$ y=\arccos \left(e^{x}\right) $$

Step-by-Step Solution

Answer

\(-\frac{e^x}{\sqrt{1-e^{2x}}}\)

Step 1: Identify the Chain Rule

The function given is a composite function: \(y = \arccos(u)\) where \(u = e^x\). We need to apply the chain rule for differentiation, which states that the derivative of \(y\) with respect to \(x\) is the derivative of \(y\) with respect to \(u\) times the derivative of \(u\) with respect to \(x\).

Step 2: Differentiate the Outer Function

The corresponding outer function is \(\arccos(u)\). The derivative of \(\arccos(u)\) with respect to \(u\) is \(-\frac{1}{\sqrt{1 - u^2}}\).

Step 3: Differentiate the Inner Function

Now differentiate the inner function: \(u = e^x\). The derivative of \(e^x\) with respect to \(x\) is \(e^x\).

Step 4: Apply the Chain Rule

Multiply the derivatives obtained from Steps 2 and 3 using the chain rule: \[\frac{dy}{dx} = \left(-\frac{1}{\sqrt{1-(e^x)^2}}\right) \cdot e^x = -\frac{e^x}{\sqrt{1-e^{2x}}}\]

Step 5: Finalize the Solution

The derivative of \(y = \arccos(e^x)\) with respect to \(x\) is thus: \[-\frac{e^x}{\sqrt{1-e^{2x}}}\]

Key Concepts

Chain Rule

The chain rule is a fundamental tool in calculus for differentiating composite functions.

• It states that if a function, say, \( y \), is dependent on another function \( u \), which in turn depends on \( x \), then the derivative of \( y \) with respect to \( x \) can be found by multiplying the derivative of \( y \) with respect to \( u \) by the derivative of \( u \) with respect to \( x \).
• This formula can be expressed mathematically as: \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \).

This is particularly useful when dealing with nested functions. By applying the chain rule, we can systematically break down and process these layers. Take our example of \( y = \arccos(e^x) \): we first identify \( y = \arccos(u) \) and \( u = e^x \). By isolating each layer, we can then differentiate them step by step.

Composite Functions

Composite functions are essentially functions made by combining two or more simpler functions.

• This means using one function's output as the input for another. It's like putting together blocks to form a more complex structure.
• Mathematically, if we have two functions \( f(x) \) and \( g(x) \), a composite function can be expressed as \( f(g(x)) \).

The differentiation process involves understanding how each part interacts and affects the whole. For instance, in the function \( y = \arccos(e^x) \), the output of the exponential function \( e^x \) directly affects the input of the inverse cosine function \( \arccos(u) \). By breaking these steps apart, we can see clearly how to apply the chain rule effectively.

Derivative of Exponential Functions

Exponential functions frequently appear in calculus, and their properties make them unique.

• The function \( e^x \), specifically, has the handy property where its derivative is itself, \( \frac{d}{dx} e^x = e^x \).
• This arises because the continuous growth rate of \( e^x \) makes it a natural candidate for describing systems that grow at a rate proportional to their current size.

In our example, the derivative of the inner function \( u = e^x \) is simply \( e^x \), which makes calculations involving exponential functions more straightforward when applying the chain rule. Recognizing these patterns can simplify and speed up your work with composite functions.

Inverse Trigonometric Functions Differentiation

Inverse trigonometric functions, like \( \arccos \), have specific differentiation rules that can be quite distinct.

• For \( \arccos(u) \), the derivative is \( -\frac{1}{\sqrt{1-u^2}} \).
• This rule reflects the rate of change of angle with respect to a change in its cosine value.

When applied in our composite function example \( y = \arccos(e^x) \), you begin by differentiating \( \arccos(u) \) with respect to \( u \). Then, make sure to include the derivative of the internal function \( u = e^x \) to properly apply the chain rule. Understanding the basic derivatives of inverse trigonometric functions allows you to approach complex composite functions with confidence.