Problem 14 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Differentiate the given function. $$f(x)=\tan \left(3 x^{2}+1\right)$$

Step-by-Step Solution

Answer

The derivative of \( f(x) = \tan(3x^2 + 1) \) is \( f'(x) = 6x \text{sec}^2(3x^2 + 1) \).

Step 1: Identify the outer and inner functions

The function is a composition of two functions. The outer function is the tangent function, and the inner function is the quadratic expression. We can write this as: \( f(x) = \tan(u) \) where \( u = 3x^2 + 1 \).

Step 2: Differentiate the outer function

Apply the chain rule to differentiate the outer function with respect to the inner function. The derivative of \( \tan(u) \) with respect to \( u \) is \( \frac{d}{du} (\tan(u)) = \frac{1}{\tan^2 u} + 1 = \text{sec}^2 (u) \).

Step 3: Differentiate the inner function

Next, find the derivative of the inner function \( u = 3x^2 + 1 \) with respect to \( x \). The derivative is \( \frac{d}{dx} (3x^2 + 1) = 6x \).

Step 4: Apply the chain rule

Combine the derivatives using the chain rule: \( \frac{df}{dx} = \frac{d}{dx}(\tan(3x^2 + 1)) = \text{sec}^2(3x^2 + 1) \times 6x \).

Step 5: Write the final answer

Therefore, the derivative of \( f(x) = \tan(3x^2 + 1) \) is \( f'(x) = 6x \text{sec}^2(3x^2 + 1) \).

Key Concepts

Chain Rule

The chain rule is a key tool when differentiating composite functions, which are functions of functions. If you have a function in the form of \(f(g(x))\), the chain rule helps differentiate it by breaking it into steps.
To use the chain rule, you:

• Differentiation of each individual function involved.
• Multiplication of those derivatives.

First, differentiate the outer function while keeping the inner function intact, and then multiply it by the derivative of the inner function.
For example, with our exercise \(f(x) = \tan(3x^2 + 1)\), we first identify the outer function \( \tan(u) \) and the inner function \( u = 3x^2 + 1 \). We then differentiate both and apply the chain rule.

tan Function Derivative

Derivatives of trigonometric functions are essential in calculus. Here, we have to differentiate \( \tan(u)\).
The derivative of the tangent function is expressed through another function: the secant function.
The rule to remember is:
\[ \frac{d}{du}(\tan(u)) = \sec^2(u). \]
This derivative reflects the rate of change of the tangent function. For our problem, when we apply the derivative to \( \tan(3x^2 + 1) \), it becomes \( \sec^2(3x^2 + 1) \).

Inner and Outer Functions

Understanding inner and outer functions is crucial for effectively applying the chain rule.
An outer function is the overarching function in a composition, and the inner function is the function inside it. In \(f(x) = \tan(3x^2 + 1)\), the outer function is \( \tan(u) \) and the inner is \( 3x^2 + 1 \).
When differentiating:

• Differentiation of the outer function \( \tan(u) \) with respect to \( u \)
• Differentiation of the inner function \( 3x^2 + 1 \) with respect to \( x \)

Combining both gives the final answer: \[ f'(x) = 6x \sec^2(3x^2 + 1). \]