Problem 11 - Calculus for Biology and Medicine

Question

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=\tan (4 x) $$

Step-by-Step Solution

Answer

The derivative of \(f(x) = \tan(4x)\) with respect to \(x\) is \(4 \sec^2(4x)\).

Step 1: Identify the Function and Its Component

The function given is \( f(x) = \tan(4x) \). Here, \(4x\) is the inside function, and \(\tan(u)\) where \(u = 4x\) is the outside function.

Step 2: Differentiate the Outside Function

The derivative of \(\tan(u)\) with respect to \(u\) is \( \sec^2(u) \). Therefore, the derivative of the outside function with respect to \(u\) is \( \sec^2(4x) \).

Step 3: Differentiate the Inside Function

Now, differentiate the inside function \(4x\) with respect to \(x\). The derivative is \(4\).

Step 4: Apply the Chain Rule

According to the chain rule, the derivative of \( \tan(4x) \) with respect to \(x\) is the derivative of the outside function times the derivative of the inside function. Thus, it is \( \sec^2(4x) \times 4 \).

Step 5: Simplify the Expression

The final expression for the derivative is \(4 \sec^2(4x)\).

Key Concepts

Understanding Derivatives in Calculus

In calculus, a derivative represents the rate at which a function is changing at any given point. It's like asking, "How fast is something happening?" For a function of a variable, the derivative tells us how the output changes as the input changes.
When working with derivatives, we often use the notation \( f'(x) \) or \( \frac{df}{dx} \) to denote the derivative of a function \( f \) with respect to \( x \). This concept is crucial in many fields, like physics for understanding motion or economics for examining changes in cost.
For example, if you have a simple function like \( f(x) = x^2 \), the derivative \( f'(x) \) is \( 2x \). This tells us that as \( x \) changes, the function \( f(x) \) changes at a rate of \( 2x \). More complex functions require additional rules, like the chain rule, which we'll explore next.

The Chain Rule: A Key Tool for Differentiation

The chain rule in calculus is a fundamental technique used to find the derivative of composite functions. A composite function is simply a function inside another function, like \( \tan(4x) \). The inside function here is \( 4x \), and the outside function is \( \tan(u) \) with \( u = 4x \).To apply the chain rule, we first differentiate the outside function, \( \tan(u) \), with respect to \( u \). The result is \( \sec^2(u) \). Next, we differentiate the inside function, \( 4x \), with respect to \( x \). The derivative is \( 4 \).
The chain rule tells us to multiply these derivatives together to find the derivative of the entire composite function. So, for \( \tan(4x) \), the chain rule gives us \( 4 \sec^2(4x) \). This approach allows us to tackle derivatives of more complex functions efficiently.
Using the chain rule is like breaking the problem into simpler parts, solving each one, and then combining the results to get the final answer.

Trigonometric Functions and Their Derivatives

Trigonometric functions, like sine, cosine, and tangent, are often used in calculus and require specific techniques for differentiation. Familiarity with these functions and their derivatives is essential for solving many calculus problems involving angles or periodic phenomena.For instance, the derivative of \( \sin(x) \) is \( \cos(x) \), and the derivative of \( \cos(x) \) is \( -\sin(x) \). The derivative of \( \tan(x) \), which is involved in our initial exercise, is \( \sec^2(x) \).
Trigonometric derivatives are particularly useful in physics for modeling waves and oscillations, like sound and light waves. In our exercise, understanding that the derivative of \( \tan(x) \) is \( \sec^2(x) \) allowed us to find the derivative of the more complex trigonometric function, \( \tan(4x) \), using the chain rule.
By mastering the derivatives of basic trigonometric functions, you set a solid foundation for handling more elaborate calculus problems involving these functions.