Problem 8 - Calculus Single Variable

Question

Calculate the derivative of the given expression with respect to \(x\). $$ \tan (\sin (x)) $$

Step-by-Step Solution

Answer

The derivative of \( \tan(\sin(x)) \) is \( \sec^2(\sin(x)) \cdot \cos(x) \).

Step 1: Recognize the Composition

The function to differentiate is a composition of two functions: the outer function is \( \tan(u) \), where \( u = \sin(x) \), and the inner function is \( \sin(x) \). This structure indicates that we will need to apply the chain rule.

Step 2: Apply the Chain Rule

The chain rule states that the derivative of a composition \( F(G(x)) \) is \( F'(G(x)) \cdot G'(x) \). Here, set \( F(u) = \tan(u) \) and \( G(x) = \sin(x) \). We need to find \( F'(u) \) and \( G'(x) \).

Step 3: Differentiate the Outer Function

Differentiate \( F(u) = \tan(u) \). The derivative is \( F'(u) = \sec^2(u) \). Substitute back \( u = \sin(x) \), which gives \( \sec^2(\sin(x)) \).

Step 4: Differentiate the Inner Function

The derivative of \( G(x) = \sin(x) \) with respect to \( x \) is \( \cos(x) \).

Step 5: Combine the Derivatives

Using the chain rule, multiply the derivative of the outer function by the derivative of the inner function: \( F'(u) \cdot G'(x) = \sec^2(\sin(x)) \cdot \cos(x) \).

Step 6: Write the Final Derivative

The final derivative of \( \tan(\sin(x)) \) with respect to \( x \) is \( \sec^2(\sin(x)) \cdot \cos(x) \).

Key Concepts

Understanding the Chain Rule

The chain rule is a fundamental concept in calculus that allows us to differentiate composite functions. When dealing with functions where one function is applied to another, like in our example, the chain rule is our go-to tool.
The idea behind the chain rule is to take the derivative of the outer function while keeping the inner function intact, and then multiply it by the derivative of the inner function.
This approach lets us smoothly handle layers of functions to find the overall derivative. In mathematical terms, if we have two functions, say \( F(u) \) and \( G(x) \), where \( u = G(x) \), then the derivative \( \frac{d}{dx}[F(G(x))] \) can be expressed as \( F'(G(x)) \cdot G'(x) \).

• The key takeaway is recognizing the composition of functions.
• Identify which is the outer function and which is the inner function.
• Focus on the derivatives of both functions and then combine them as per the chain rule.

This technique is essential for calculus problems involving nested functions.

Exploring Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are not just pivotal in geometry but also play a significant role in calculus.
For our exercise, we dealt with the sine function and its relationship within a tangent function. Here's a quick recap of what these functions are about:

• The sine function, \( \sin(x) \), takes an angle and returns the ratio of the side opposite the angle to the hypotenuse in a right triangle.
• The tangent function, \( \tan(x) \), gives us the ratio of the sine of an angle to the cosine of that angle, often visualized as the slope of the line in the unit circle.
• When differentiating \( \tan(x) \), its derivative is \( \sec^2(x) \).

In the exercise, knowing these properties helped us apply the chain rule correctly by identifying and differentiating these functions accurately. The derivatives exhibit certain patterns, such as \( \sin'(x) = \cos(x) \), which are essential building blocks for more complex calculus operations.

Composition of Functions Simplified

Composition of functions is about layering functions within each other. Think of it as creating a compound operation from simpler ones.
For instance, with our original function \( \tan(\sin(x)) \), \( \tan \) is applied to the result of \( \sin(x) \). This nesting of functions is what necessitates the use of the chain rule.
Recognizing composition:

• Identify the outer and inner functions.
• Understand how one function flows into the other, creating layers.
• Separate the outer from the inner when using calculus methods like differentiation.

By treating the composition as a hierarchy of functions, you can systematically differentiate using the chain rule. This understanding is foundational in calculus, especially when you work with more intricate functions that interlock in similar manners. Mastering this lets you tackle complex derivatives by breaking them into manageable parts.