Problem 42 - Calculus - AP Edition

Question

Use a CAS to find dy/dx. $$y=\tan ^{4}\left(2+\frac{(7-x) \sqrt{3 x^{2}+5}}{x^{3}+\sin x}\right)$$

Step-by-Step Solution

Answer

Use chain and quotient rules, then simplify using CAS.

Step 1: Understand the Problem

We are given the function \( y = ( \tan \left( 2 + \frac{ (7-x) \sqrt{3x^2 + 5} }{ x^3 + \sin(x) } \right) )^4 \) and are tasked with finding \( \frac{dy}{dx} \), the derivative of \( y \) with respect to \( x \).

Step 2: Apply the Chain Rule

The function \( y = (\tan(u(x)))^4 \) where \( u(x) = 2 + \frac{ (7-x) \sqrt{3x^2 + 5} }{ x^3 + \sin(x) } \). The chain rule states that \( \frac{dy}{dx} = 4 \cdot (\tan(u))^3 \cdot \sec^2(u) \cdot \frac{du}{dx} \).

Step 3: Differentiate Inside Function using Quotient Rule

To find \( \frac{du}{dx} \), apply the quotient rule: \( \frac{d}{dx} \left( \frac{(7-x) \sqrt{3x^2 + 5}}{x^3 + \sin(x)} \right) = \frac{ (x^3 + \sin(x)) \cdot \frac{d}{dx}((7-x) \sqrt{3x^2 + 5}) - (7-x) \sqrt{3x^2 + 5} \cdot (3x^2 + \cos(x))}{(x^3 + \sin(x))^2} \).

Step 4: Differentiate the Numerator

For the numerator \((7-x) \sqrt{3x^2 + 5}\), use the product rule: \( \frac{d}{dx}((7-x) \cdot \sqrt{3x^2 + 5}) = -1 \cdot \sqrt{3x^2 + 5} + (7-x) \cdot \frac{1}{2\sqrt{3x^2 + 5}} \cdot 6x \).

Step 5: Simplify and Substitute

Simplify the expression for \( \frac{du}{dx} \) obtained from above and substitute it into the chain rule equation from Step 2: \( \frac{dy}{dx} = 4 \cdot (\tan(u))^3 \cdot \sec^2(u) \cdot \frac{du}{dx} \).

Step 6: Use Computational Algebra System (CAS)

Input the resulting expression into a CAS to simplify the computations and obtain the derivative. CAS tools are perfect for handling complex symbolic derivatives efficiently.

Key Concepts

Chain Rule

When dealing with composite functions, where one function is nested inside another, the chain rule comes in handy. It helps us differentiate such functions by systematically breaking them down. In our problem, \( y = ( \tan(u(x)))^4 \), involves nesting because the tangent function and the power of four are applied to \( u(x) \). Here’s how the chain rule works:

• First, differentiate the outer function. In this case, the outer function is \( (\tan(u))^4 \). Using the power rule, this becomes \( 4(\tan(u))^3 \).
• Then, differentiate the inner function, which is \( \tan(u) \). The derivative of \( \tan(u) \) is \( \sec^2(u) \).
• Finally, multiply these results by the derivative of \( u(x) \), which requires separate computation.

Applying the chain rule requires practice to identify every function layer. Always work step by step. Being diligent makes complex derivatives manageable.

Quotient Rule

In calculus, whenever you have a function that is the ratio of two functions, the quotient rule is the tool you need. In our exercise, function \( u(x) \) is a ratio: \( u(x) = 2 + \frac{ (7-x) \sqrt{3x^2 + 5} }{ x^3 + \sin(x) } \). To find the derivative of such an expression, use the quotient rule.This rule states that for two functions \( f(x) \) and \( g(x) \), the derivative of \( \frac{f}{g} \) is given by:\[\frac{d}{dx} \left( \frac{f}{g} \right) = \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2}.\]

• Identify the numerator function \( f(x) \) and the denominator \( g(x) \).
• Differentiate both \( f(x) \) and \( g(x) \).
• Apply the formula to combine these derivatives.

Mastering the quotient rule helps tackle rational derivatives efficiently. Calculations require careful simplification to prevent errors.

Product Rule

The product rule is used when a function is the product of two other functions. This rule allows us to differentiate products efficiently. In our exercise, the numerator of \( u(x) \), \((7-x) \sqrt{3x^2 + 5}\), is a product of two functions, \( (7-x) \) and \( \sqrt{3x^2 + 5} \).Here’s the plan with the product rule:

• For functions \( f(x) \) and \( g(x) \), the derivative is \( f'(x)g(x) + f(x)g'(x) \).
• Differentiate each function separately. For example, the derivative of \( (7-x) \) is \(-1\).
• And for \( \sqrt{3x^2 + 5} \), use the power rule. Simplifying gives \( \frac{1}{2\sqrt{3x^2 + 5}} \cdot 6x \).
• Add results according to the formula.

By breaking tasks into smaller parts, the product rule confidently handles derivatives of multiplied functions.

Computational Algebra System (CAS)

A Computational Algebra System (CAS) is software that can perform symbolic mathematics. This means it can solve problems like simplifying derivatives, integrals, and algebraic expressions, handling variables symbolically rather than as numbers. Examples of CAS tools include Mathematica, Maple, and Maxima.In complex calculus problems like ours, CAS becomes indispensable because:

• It automates complex derivative solutions, minimizing manual error risk.
• It visualizes functions and derivatives, providing a better understanding of behavior and properties.
• CAS can handle multiple and intricate symbolic computations swiftly, as seen when we reduce our derivative \( \frac{dy}{dx} \) to a simplified form.

Using a CAS doesn’t replace learning. Instead, it complementarily enhances understanding, enabling survival in tackling intricate mathematical challenges in both academic and professional settings.