Problem 45 - Calculus for Biology and Medicine

Question

Differentiate the functions with respect to the independent variable. $$ f(x)=\ln \left(\tan x^{2}\right) $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = \frac{2x \sec^2(x^2)}{\tan(x^2)} \).

Step 1: Identify the function to differentiate

The function given is \[ f(x) = \ln(\tan(x^2)) \]. This is a composite function consisting of a natural logarithm and a tangent function applied to \(x^2\).

Step 2: Apply the Chain Rule

To differentiate \( f(x) = \ln(\tan(x^2)) \), we use the chain rule, which states that if you have a function of a function, such as \( g(h(x)) \), its derivative is \( g'(h(x)) \times h'(x) \). Identify \( u = \tan(x^2) \) and hence, \( f(x) = \ln(u) \).

Step 3: Differentiate the Natural Logarithm

The derivative of \( \ln(u) \) with respect to \( u \) is \( \frac{1}{u} \). So, the derivative of \( \ln(\tan(x^2)) \) is \( \frac{1}{\tan(x^2)} \) times the derivative of \( \tan(x^2) \).

Step 4: Differentiate the Tangent Function

Now, find the derivative of \( \tan(x^2) \) with respect to \( x \). The derivative of \( \tan(v) \) with respect to \( v \) is \( \sec^2(v) \). So, the derivative of \( \tan(x^2) \) with respect to \( x^2 \) is \( \sec^2(x^2) \). Then, multiply by the derivative of \( x^2 \), which is \( 2x \). Hence, the derivative is \( 2x \sec^2(x^2) \).

Step 5: Combine the Results

Now combine the results from Steps 3 and 4. The derivative of \( f(x) = \ln(\tan(x^2)) \) with respect to \( x \) is given by \[ f'(x) = \frac{1}{\tan(x^2)} \cdot 2x \sec^2(x^2) \]. Simplifying gives:\[ f'(x) = \frac{2x \sec^2(x^2)}{\tan(x^2)} \].

Key Concepts

Chain Rule

The chain rule is a very important tool in calculus when dealing with composite functions. It helps you find the derivative of a function that is made up of other functions. You can think of it as peeling back the layers of an onion, working your way from the outside layer to the inside.
If you have a composite function, which is a function within another function (let’s call them outer function \( g \) and inner function \( h \)), it is written as \( g(h(x)) \).
The chain rule formula is: \( (g \circ h)'(x) = g'(h(x)) \cdot h'(x) \). This means you first take the derivative of the outer function, keeping the inner function as it is, and then multiply it by the derivative of the inner function.

• Identify the inner and outer functions
• Differentiate each function separately
• Multiply them together as per the chain rule

When you practice this step by step, it makes finding derivatives much simpler, especially when working with complex functions.

Composite Function

A composite function is basically a plug-in of one function into another. It's like nesting functions within each other.

For example, if you have two functions \( g(x) \) and \( h(x) \), then the composition is \( g(h(x)) \). You are essentially substituting \( h(x) \) everywhere there's \( g(x) \). This is key to solving problems involving the chain rule, as it connects the inner and outer functions, forming a pathway to following through the differentiation processes step by step.

• Understand the order: which function is inside and which is outside
• Know how to "navigate" through these layers

With clear identification of which function is which, working through derivatives becomes far more manageable.

Natural Logarithm

The natural logarithm, often denoted as \( \ln \), is the logarithm to the base \( e \) where \( e \approx 2.718 \). It is common in calculus due to its unique properties and relations to exponential functions.

One of its key characteristics is how it simplifies differentiation. If \( u(x) \) is your function, then \( \ln(u(x)) \) differentiates to \( \frac{1}{u(x)} \cdot u'(x) \). The beauty of the natural logarithm is that it turns multiplication in differentiation into addition, and division into subtraction.

Here’s what you get with natural logarithms:

• The derivative of \( \ln u \) is \( \frac{1}{u} \times \) (the derivative of \( u \))
• They often help in simplifying complex derivatives

The natural logarithm helps simplify chain rule problems significantly when dealing with products, quotients, and powers of \( e \).