Question

Use logarithmic differentiation to find \(\frac{d y}{d x}\). $$ y=(\sin 2 x)^{4 x} $$

Short answer

\(\frac{dy}{dx} = (\sin 2x)^{4x} \left(4 \ln(\sin 2x) + \frac{8x \cos 2x}{\sin 2x}\right)\).

Step-by-step solution

Take the Natural Logarithm

Start by taking the natural logarithm of both sides of the equation. This helps in simplifying the power and product forms.\[\ln y = \ln \left( (\sin 2x)^{4x} \right)\]

Apply Logarithmic Identity

Apply the power rule of logarithms, which states that \(\ln(a^b) = b \cdot \ln(a)\).\[\ln y = 4x \cdot \ln(\sin 2x)\]

Differentiate Both Sides with Respect to x

Differentiate implicitly with respect to \(x\). Remember to use the product rule for \(4x \cdot \ln(\sin 2x)\).\[\frac{1}{y} \cdot \frac{dy}{dx} = 4 \cdot \ln(\sin 2x) + 4x \cdot \frac{1}{\sin 2x} \cdot (\cos 2x \cdot 2)\]

Simplify the Derivative Expression

Simplify the derivative expression obtained in Step 3.\[\frac{1}{y} \cdot \frac{dy}{dx} = 4 \ln(\sin 2x) + \frac{8x \cos 2x}{\sin 2x}\]

Solve for dy/dx

Multiply both sides by \(y\) to solve for \(\frac{dy}{dx}\) using the original \(y\) value \((\sin 2x)^{4x})\).\[\frac{dy}{dx} = y \left(4 \ln(\sin 2x) + \frac{8x \cos 2x}{\sin 2x}\right)\]Substitute \(y = (\sin 2x)^{4x}\) back into the equation:\[\frac{dy}{dx} = (\sin 2x)^{4x} \left(4 \ln(\sin 2x) + \frac{8x \cos 2x}{\sin 2x}\right)\]

Key concepts

Implicit Differentiation

Implicit differentiation is a useful technique when dealing with equations where variables cannot be easily separated. For functions defined implicitly, one differentiates both sides of the equation with respect to a specific variable, often without solving for one variable in terms of the other beforehand. This method is particularly handy when you're given relationships like the one \( y = (\sin 2x)^{4x} \), where isolating \( y \) could be cumbersome.
When taking the derivative of both sides of this equation, the chain rule comes into play. You differentiate \( \ln y = 4x \cdot \ln(\sin 2x) \) with respect to \( x \) by implicit differentiation, treating \( y \) as a function of \( x \). This leads to the derivative of the natural log of \( y \), which is \( \frac{1}{y} \cdot \frac{dy}{dx} \), on one side and the application of derivatives to the expression on the other side. It allows you to solve for \( \frac{dy}{dx} \) without explicitly needing \( y \) alone, which simplifies the process.

Power Rule of Logarithms

The power rule of logarithms is a critical tool in logarithmic differentiation. It simplifies handling expressions that involve a term raised to a power. The rule states that \( \ln(a^b) = b \cdot \ln(a) \). This principle helps us break down complex exponential expressions into products, which are easier to differentiate.
For instance, in the equation \( \ln y = \ln((\sin 2x)^{4x}) \), applying the power rule transforms it to \( \ln y = 4x \cdot \ln(\sin 2x) \). By moving the exponent down as a coefficient, the expression becomes more manageable. The power rule simplifies our differentiation task by helping to rotate between forms that are easier to handle mathematically.

Product Rule

The product rule is essential when differentiating products of two functions. When faced with a function like \( 4x \cdot \ln(\sin 2x) \), you need to consider the interaction between these parts of the function. The product rule tells us that if we have a product \( u \cdot v \), its derivative is \( u'v + uv' \).
Applying this to our function, identify \( u = 4x \) and \( v = \ln(\sin 2x) \). Then, differentiate each part:

• \( u' = 4 \)
• \( v' = \frac{d}{dx}(\ln(\sin 2x)) = \frac{1}{\sin 2x} \cdot \cos 2x \cdot 2 \)

Consequently, the derivative \( \frac{d}{dx}(4x \cdot \ln(\sin 2x)) = 4 \cdot \ln(\sin 2x) + 8x \cos 2x / \sin 2x \).
Hence, the product rule enables us to find the derivative of more complex expressions by breaking them into manageable parts.

Natural Logarithm

Natural logarithms (ln) are logarithms with the base \( e \), an important constant approximately equal to 2.718. It's often used in calculus due to its unique properties that simplify differentiation and integration processes.
When tackling logarithmic differentiation, the natural logarithm plays a crucial part because it naturally aligns with exponential growth and decay models common in mathematical functions.
In our exercise, we initially transform the equation by applying the natural logarithm to simplify the expression: \( \ln y = \ln((\sin 2x)^{4x}) \). This application enables us to apply the power and product rules effectively, as seen in the steps: the logarithm simplifies exponentiation, and transforms intricate expressions into sums and products, which are simpler to differentiate. Thus, natural logarithms are integral to solving complex differentiation problems efficiently.