Question

Logarithmic differentiation Use logarithmic differentiation to evaluate \(f^{\prime}(x)\). $$f(x)=\frac{\tan ^{10} x}{(5 x+3)^{6}}$$

Short answer

Question: Find the derivative of the function \(f(x) = \frac{\tan^{10}(x)}{(5x+3)^6}\) using logarithmic differentiation. Answer: The derivative of the function \(f(x)\) with respect to \(x\) is: $$f^{\prime}(x) = \frac{10\tan^{9}(x) \cdot (\sec^2 x)(5x+3)^6 - 30 \tan^{10}(x)(5x+3)^5}{(5x+3)^{12}}$$

Step-by-step solution

Take the natural logarithm of the function

Find the natural logarithm of \(f(x)\). $$\ln{f(x)} = \ln\left(\frac{\tan^{10} x}{(5x+3)^6}\right)$$

Apply logarithm properties

Use the logarithm properties to simplify the expression. $$\ln{f(x)} = 10\ln{(\tan x)} - 6\ln{(5x+3)}$$

Differentiate with respect to x

Differentiate the expression with respect to \(x\). $$\frac{f^{\prime}(x)}{f(x)} = 10\left[\frac{1}{\tan x} \cdot (\sec^2 x)\right] - 6\left[\frac{5}{5x+3}\right]$$

Multiply by \(f(x)\) and simplify

Since we require the expression of \(f^{\prime}(x)\), multiply both sides of the equation by \(f(x)\) and simplify the expression. $$f^{\prime}(x) = f(x) \left[ 10\left[\frac{1}{\tan x} \cdot (\sec^2 x)\right] - 6\left[\frac{5}{5x+3}\right] \right]$$ Using the given function \(f(x)\), $$f^{\prime}(x) = \frac{\tan ^{10} x}{(5 x+3)^{6}} \left[ 10\left[\frac{1}{\tan x} \cdot (\sec^2 x)\right] - 6\left[\frac{5}{5x+3}\right] \right]$$

Final answer

Combine and simplify the terms to obtain the final derivative function. $$f^{\prime}(x) = \frac{10\tan^{9}(x) \cdot (\sec^2 x)(5x+3)^6 - 30 \tan^{10}(x)(5x+3)^5}{(5x+3)^{12}}$$

Key concepts

Derivative

A derivative represents the rate at which a function is changing at any given point. It is a fundamental concept in calculus, used to find the slope of the tangent line to the curve of a function. In simple terms, it tells us how a function behaves as its input changes.

When using logarithmic differentiation to find the derivative of a complex function like\( f(x) = \frac{\tan^{10} x}{(5x+3)^6} \), we take advantage of the unique properties of logarithms. This method allows for easier differentiation of products, quotients, and powers.

• First, take the natural logarithm of both sides.
• Then, apply properties of logarithms to simplify the expression.
• Differentiate the resulting expression with respect to \( x \).
• After differentiation, solve for the derivative \( f'(x) \).

Logarithmic differentiation makes some seemingly difficult derivative calculations more manageable.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is the logarithm to the base \( e \), where \( e \approx 2.71828 \). It is particularly useful in calculus because of its unique derivation properties.

Natural logarithms help transform complex functions involving products and powers into simpler forms, making them easier to differentiate. For a function \( f(x) = \frac{\tan^{10} x}{(5x+3)^6} \), the natural logarithm simplifies the differentiation process.

• The property \( \ln(ab) = \ln(a) + \ln(b) \) splits products into sums.
• The property \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \) converts divisions into subtractions.
• The power rule \( \ln(a^b) = b\ln(a) \) helps manage exponents.

In our specific example, the natural logarithm allows us to separate the power of the tangent and the polynomial term, simplifying the function dramatically.

Trigonometric Functions

Trigonometric functions appear frequently in calculus, and their derivatives are fundamental to solving many problems. These functions include \( \sin(x), \cos(x), \tan(x) \), and their respective reciprocal and inverse functions.

In the example \( f(x) = \frac{\tan^{10} x}{(5x+3)^6} \), the function \( \tan(x) \) plays a significant role. The derivative of \( \tan(x) \) is \( \sec^2(x) \), which becomes crucial during differentiation.

• First, recognize the trigonometric function's involvement in the given function.
• Application of the chain rule: When differentiating \( \tan^{10}(x) \), consider the power as well.

By incorporating these derivatives, the function simplifies, allowing for a smoother process to find \( f'(x) \). The interplay of these functions showcases the preciseness and technique required in calculus.

Simplification

Simplification is about reducing a mathematical expression to its simplest form without changing its value, making it easier to interpret and work with.

Once you have applied logarithmic differentiation and taken the derivatives, simplifying the expression is key to obtaining the final derivative form. Let's see how:

• Breakdown complex expressions into familiar components to reduce error.
• Combine like terms and use algebraic manipulation to condense expressions.

In the original exercise, simplification led from a complex derivative to a more understandable expression:
\[ f'(x) = \frac{10\tan^9(x) \cdot (\sec^2 x)(5x+3)^6 - 30 \tan^{10}(x)(5x+3)^5}{(5x+3)^{12}} \]
This form is essential for analysis and further calculation, showcasing the importance of simplification in finding accurate derivatives.