Question

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 given function \(f(x) = \frac{\tan ^{10} x}{(5 x+3)^{6}}\) using logarithmic differentiation. Answer: \(f'(x) = 10\tan^9 x\sec^2 x - 30\frac{\tan^{10}x}{(5x+3)^7}\)

Step-by-step solution

Take the natural logarithm of both sides

\(\ln(f(x)) = \ln\left(\frac{\tan ^{10} x}{(5 x+3)^{6}}\right)\).

Simplify the expression using logarithm properties

Use the properties of logarithms to simplify the right-hand side of the equation: \(\ln(f(x)) = 10\ln(\tan x) - 6\ln(5x+3)\).

Differentiate both sides of the equation with respect to x

Now, we'll differentiate both sides of the equation with respect to x: \(\frac{f'(x)}{f(x)} = 10\frac{d}{dx}(\ln(\tan x)) - 6\frac{d}{dx}(\ln(5x+3))\) Applying the chain rule, we get: \(\frac{f'(x)}{f(x)} = 10\frac{1}{\tan x}\cdot\frac{d}{dx}(\tan x) - 6\frac{1}{5x+3}\cdot\frac{d}{dx}(5x+3)\) \(\frac{f'(x)}{f(x)} = 10\frac{1}{\tan x}\cdot\sec^2 x - 6\frac{1}{5x+3}\cdot 5\)

Solve for \(f'(x)\), the derivative of the function

Now we can solve for \(f'(x)\): \(f'(x) = f(x)\cdot \left(10\frac{1}{\tan x}\cdot\sec^2 x - 6\frac{1}{5x+3}\cdot 5\right)\) We know that \(f(x) = \frac{\tan ^{10} x}{(5 x+3)^{6}}\). By multiplying both sides by the simplified expression, we get: \(f'(x) = \frac{\tan ^{10} x}{(5 x+3)^{6}}\cdot \left(10\frac{1}{\tan x}\cdot\sec^2 x - 6\frac{1}{5x+3}\cdot 5\right)\) Now simplify the expression: \(f'(x) = 10\tan^9 x\sec^2 x - 30\frac{\tan^{10}x}{(5x+3)^7}\) So the derivative of the given function is: \(f'(x) = 10\tan^9 x\sec^2 x - 30\frac{\tan^{10}x}{(5x+3)^7}\)

Key concepts

Chain Rule

The chain rule is a fundamental tool in calculus for differentiating composite functions. When you have a function nested inside another function, the chain rule helps you find the derivative. It states that if you have a composite function \( y = f(g(x)) \), then its derivative is the product of the derivative of the outer function and the derivative of the inner function. This can be mathematically represented as \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).

In the context of our exercise, when differentiating expressions such as \( \ln(\tan x) \), we apply the chain rule. The outer function here is the natural logarithm, and the inner function is \( \tan x \). Thus, the derivative \( \frac{d}{dx}(\ln(\tan x)) \) becomes \( \frac{1}{\tan x} \cdot \sec^2 x \). Similarly, for \( \ln(5x+3) \), the derivative is \( \frac{1}{5x+3} \cdot 5 \).

Applying the chain rule properly ensures accurate results especially in complex expressions involving multiple functions.

Derivative

A derivative represents the rate at which a function changes with respect to a variable. It's a fundamental concept in calculus, symbolized as \( f'(x) \) when differentiating a function \( f(x) \).

• The derivative of a constant is zero.
• The derivative of \( x^n \) is \( nx^{n-1} \).
• The derivative of \( \sin x \) is \( \cos x \), and of \( \tan x \) is \( \sec^2 x \).

In logarithmic differentiation, we use the derivative of logarithmic functions, notably: \( \frac{d}{dx}(\ln x) = \frac{1}{x} \).

Our problem involves differentiating a quotient \( \frac{\tan^{10} x}{(5x+3)^6} \). By taking the natural logarithm, simplifying using properties, and differentiating, we manage more complex differentiation using related derivatives like \( \tan x \) and the constants involved in polynomial terms. This approach breaks down the derivative into manageable parts, which are then combined for the final solution.

Logarithmic Properties

Logarithmic properties are crucial in simplifying complex expressions. Using logarithms in differentiation makes handling products, quotients, and powers more manageable.

• The logarithm of a product is the sum of the logarithms: \( \ln(ab) = \ln a + \ln b \).
• The logarithm of a quotient is the difference of the logarithms: \( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \).
• The logarithm of a power brings down the exponent: \( \ln(a^b) = b\ln a \).

For our exercise, we first take the natural logarithm of both sides to leverage these properties. The function \( \frac{\tan^{10}x}{(5x+3)^6} \) becomes \( 10\ln(\tan x) - 6\ln(5x+3) \) after applying the properties. These simplified terms are easier to differentiate logarithmically, transforming a complex quotient into a manageable subtraction of two terms.