Problem 41 - Calculus

Question

Evaluate by any method. $$ \frac{d}{d x} \ln (\sec x+\tan x) $$

Step-by-Step Solution

Answer

The derivative is \( \frac{\sec x \tan x + \sec^2 x}{\sec x + \tan x} \).

Step 1: Identify the function to differentiate

The problem asks us to differentiate the function \( \ln(\sec x + \tan x) \) with respect to \( x \). We will use the chain rule for differentiation.

Step 2: Use the Chain Rule

The chain rule states that if you have a composite function \( f(g(x)) \), its derivative is \( f'(g(x)) \cdot g'(x) \). Here, \( f(u) = \ln(u) \) and \( u = \sec x + \tan x \). The derivative of \( f(u) \) with respect to \( u \) is \( \frac{1}{u} \). We must now find \( \frac{du}{dx} \).

Step 3: Differentiate \( \sec x + \tan x \)

The derivative of \( \sec x \) is \( \sec x \tan x \), and the derivative of \( \tan x \) is \( \sec^2 x \). Therefore, \( \frac{du}{dx} = \sec x \tan x + \sec^2 x \).

Step 4: Combine the Derivatives

Substitute back into the chain rule formula. The derivative of \( \ln(\sec x + \tan x) \) is obtained by multiplying \( \frac{1}{\sec x + \tan x} \) by \( \sec x \tan x + \sec^2 x \). Thus, the derivative is \[ \frac{\sec x \tan x + \sec^2 x}{\sec x + \tan x}. \]

Key Concepts

Chain Rule

The Chain Rule is a vital differentiation technique in calculus, especially when dealing with composite functions. This rule helps us differentiate a function that is nested within another function.

Imagine you have a function like \( f(g(x)) \). The Chain Rule states that the derivative of this composite function is \( f'(g(x)) \times g'(x) \). Essentially, you first differentiate the outer function, and then multiply it by the derivative of the inner function.

For instance, consider \( f(u) = \ln(u) \) and \( u = \sec x + \tan x \). When using the Chain Rule, first compute the derivative \( \frac{1}{u} \) of \( \ln(u) \). Next, differentiate the inside function \( u \) to find \( g'(x) \). By multiplying these derivatives, you obtain the derivative of the composite function \( \ln(\sec x + \tan x) \). This systematic approach simplifies finding derivatives of complex expressions.

Derivative of Trigonometric Functions

Trigonometric functions often appear in calculus, and their derivatives are essential for solving many problems. Knowing these derivatives is crucial.

Two fundamental trigonometric functions that we often differentiate are \( \sec x \) and \( \tan x \). Both have straightforward derivatives:

• The derivative of \( \sec x \) is \( \sec x \tan x \).
• The derivative of \( \tan x \) is \( \sec^2 x \).

Understanding these derivatives becomes incredibly useful when dealing with trigonometric expressions. As we see in the problem, understanding the derivatives of \( \sec x \) and \( \tan x \) leads directly to finding the derivative of \( u = \sec x + \tan x \). This knowledge simplifies calculations and anchors the solution process.

Logarithmic Differentiation

Logarithmic differentiation is a special technique that comes in handy when differentiating functions involving logarithms, especially those with complex compositions.

The process starts by recognizing the structure of the function, in this case, a logarithmic form: \( \ln(\sec x + \tan x) \). The derivative of \( \ln(u) \) with respect to \( u \) is \( \frac{1}{u} \). From there, the Chain Rule lets us move forward by working on the inner function \( \sec x + \tan x \).

Coupling logarithmic differentiation with the chain rule allows one to simplify processes that otherwise seem cumbersome. It combines the derivative of natural logarithm functions with the sensitivity to modifications in the internal structure of a composite function, streamlining the process of differentiation in complex scenarios.