Problem 42 - Calculus Single Variable

Question

Differentiate the given expression with respect to \(x\). $$ \tanh (\ln (x+2)) $$

Step-by-Step Solution

Answer

The derivative is \( \frac{\text{sech}^2(\ln(x+2))}{x+2} \).

Step 1: Apply the Chain Rule

To differentiate the function \( \tanh(\ln(x+2)) \), we will use the chain rule. The chain rule states that if a function \( y = f(g(x)) \), then the derivative \( y' = f'(g(x)) \cdot g'(x) \). Let \( u = \ln(x+2) \), so our function is \( \tanh(u) \).

Step 2: Differentiate Outer Function

The derivative of \( \tanh(u) \) with respect to \( u \) is \( \text{sech}^2(u) \). Hence, \( \frac{d}{du}\left[ \tanh(u) \right] = \text{sech}^2(u) \).

Step 3: Differentiate Inner Function

The derivative of \( u = \ln(x+2) \) with respect to \( x \) is \( \frac{1}{x+2} \). Hence, \( \frac{d}{dx}\left( \ln(x+2) \right) = \frac{1}{x+2} \).

Step 4: Combine Derivatives Using Chain Rule

Now, we apply the chain rule: \( \frac{dy}{dx} = \frac{d}{du}\left[ \tanh(u)\right] \cdot \frac{d}{dx}\left( u \right) \). This becomes \( \text{sech}^2(\ln(x+2)) \cdot \frac{1}{x+2} \).

Step 5: Final Expression for the Derivative

The derivative of \( \tanh(\ln(x+2)) \) with respect to \( x \) is \( \frac{\text{sech}^2(\ln(x+2))}{x+2} \).

Key Concepts

Chain Rule

The chain rule is an essential technique in calculus, particularly useful for differentiating compositions of functions. When you have a function nested inside another, like in the expression \( \tanh(\ln(x+2)) \), understanding the chain rule is vital. It helps you differentiate each layer separately and then connect them.
Here's the fundamental idea:

• Look at your function as composed of two functions: an outer function \( f \) and an inner function \( g \).
• The outer function in our case is \( \tanh(u) \), with \( u = \ln(x+2) \) as the inner function.
• The chain rule states: The derivative of \( y = f(g(x)) \) is \( f'(g(x)) \cdot g'(x) \).

So, you first differentiate the outer function with respect to the inner function and then multiply by the derivative of the inner function with respect to \( x \). This approach turns what could be a complex problem into manageable pieces.

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions but for the hyperbola, just like trigonometric functions involve the circle.

• Common hyperbolic functions include \( \sinh, \cosh, \) and \( \tanh \).
• For our problem, the focus is on the function \( \tanh(x) \), which is the hyperbolic tangent.
• The derivative of \( \tanh(x) \) is \( \text{sech}^2(x) \), where \( \text{sech}(x) \) is the hyperbolic secant function, defined as \( \frac{1}{\cosh(x)} \).

Understanding the properties of hyperbolic functions is crucial because they frequently appear in calculus, solving differential equations, and physics. They behave similarly to their trigonometric counterparts but with unique characteristics of the hyperbola, making them fascinating to study.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a logarithm to the base \( e \), where \( e \approx 2.71828 \). It is a fundamental function in calculus that helps relate multiplicative processes to additive scales. The natural logarithm has several properties that make it especially valuable:

• The derivative of \( \ln(x) \) is \( \frac{1}{x} \), highlighting how it simplifies the differentiation of logarithmic expressions.
• In our expression \( \ln(x+2) \), the derivative utilizes this basic property but becomes \( \frac{1}{x+2} \) due to the chain rule applied to the inner function \( (x+2) \).
• The natural logarithm transforms products into sums, often simplifying the integration process.

This function often appears in contexts like exponential growth and decay, making a comprehensive understanding of its derivative critical in calculus.