Problem 38 - Calculus - AP Edition

Question

Find \(d y / d x\) using logarithmic differentiation. $$y=\frac{\sin x \cos x \tan ^{3} x}{\sqrt{x}}$$

Step-by-Step Solution

Answer

\(\frac{dy}{dx} = \frac{\sin x \cos x \tan^3 x}{\sqrt{x}} (\cot x - \tan x + 3 \sec^2 x \tan x - \frac{1}{2x})\).

Step 1: Take the Natural Logarithm of Both Sides

To apply logarithmic differentiation, start by taking the natural logarithm of both sides. This will help us simplify the differentiation process.\[ \ln y = \ln \left( \frac{\sin x \cos x \tan ^{3} x}{\sqrt{x}} \right) \]Use the property of logarithms that \( \ln \left( \frac{a}{b} \right) = \ln a - \ln b \), we write:\[ \ln y = \ln(\sin x \cos x \tan^3 x) - \ln(\sqrt{x}) \]

Step 2: Apply Logarithmic Properties

Apply the logarithmic property \( \ln(ab) = \ln a + \ln b \) to the terms inside the logarithm of the numerator:\[ \ln y = \ln(\sin x) + \ln(\cos x) + \ln(\tan^3 x) - \ln(x^{1/2}) \]Using \( \ln(a^b) = b\ln a \) for the power terms:\[ \ln y = \ln(\sin x) + \ln(\cos x) + 3\ln(\tan x) - \frac{1}{2} \ln x \]

Step 3: Differentiate Both Sides

Now, differentiate both sides with respect to \(x\). On the left, use the chain rule:\[ \frac{d}{dx}(\ln y) = \frac{1}{y} \frac{dy}{dx} \]On the right, differentiate term by term:\[ \frac{d}{dx}(\ln y) = \frac{d}{dx}(\ln(\sin x)) + \frac{d}{dx}(\ln(\cos x)) + \frac{d}{dx}(3\ln(\tan x)) - \frac{d}{dx}(\frac{1}{2}\ln x) \]This gives us:\[ \frac{1}{y} \frac{dy}{dx} = \cot x - \tan x + 3 \sec^2 x \cdot \tan x - \frac{1}{2x} \]

Step 4: Solve for \( \frac{dy}{dx} \)

Multiply through by \(y\) to solve for \( \frac{dy}{dx} \):\[ \frac{dy}{dx} = y \left( \cot x - \tan x + 3 \tan x \sec^2 x - \frac{1}{2x} \right) \]Replace \(y\) with its original value:\[ \frac{dy}{dx} = \frac{\sin x \cos x \tan^3 x}{\sqrt{x}} \left( \cot x - \tan x + 3 \sec^2 x \tan x - \frac{1}{2x} \right) \]

Step 5: Simplify the Expression (If Possible)

Check if the expression can be simplified any further. In this case, leave the expression as is for clarity.Thus, the derivative is given by:\[ \frac{dy}{dx} = \frac{\sin x \cos x \tan^3 x}{\sqrt{x}} \left( \cot x - \tan x + 3 \sec^2 x \tan x - \frac{1}{2x} \right) \]

Key Concepts

Natural Logarithm

The natural logarithm, represented as `ln`, is a mathematical function that is a particular logarithm where the base is the number `e` (approximately 2.71828). Logarithms convert multiplicative relationships into additive ones, which comes in handy for differentiation. When you take the natural logarithm of a product, such as in our exercise, it allows us to separate the components into a sum of logs using the property:

• \( \ln(ab) = \ln a + \ln b \)
• \( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \)

This simplification is crucial in making the differentiation process more manageable, especially when dealing with complex expressions. By simplifying the expression using the natural logarithm, we can differentiate term by term rather than dealing with the whole product or quotient at once.

Chain Rule in Differentiation

The chain rule is a fundamental differentiation technique used when differentiating composite functions. A composite function involves two functions where an outside function contains an inside function. When we have a function of the form \( f(g(x)) \), the chain rule states:

• \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \)

In our logarithmic differentiation process, when we took the logarithm of \( y \), we formed an implicit differentiation scenario with respect to \( y \), needing to apply the chain rule:

• If \( \ln y \) differentiates to \( \frac{1}{y} \frac{dy}{dx} \), this step utilizes the chain rule because \( y \) is a function of \( x \).

Applying the chain rule correctly allows us to find the derivative of complex functions built from simpler ones effectively. Remember, identifying when to employ the chain rule is crucial for successfully differentiating composite functions.

Trigonometric Functions Differentiation

Differentiating trigonometric functions is a common task in calculus, and each trigonometric function has its own specific derivative formula. Understanding these formulas greatly aids in differentiating expressions involving trigonometric functions:

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

In our exercise involving logarithmic differentiation, we had trigonometric terms like \( \sin x \), \( \cos x \), and \( \tan^3 x \). When differentiating \( \ln(\tan^3 x) \), for instance, we use the fact that the derivative of \( \tan x \) is \( \sec^2 x \), and multiply by 3 due to the power rule, obtaining:

• \( 3(\sec^2 x) \)

Mastering these basic derivatives is essential for tackling more complex problems—first, identify the trigonometric function, then apply its derivative rule.