Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).
f(x) = In(sec⁴x tan² x)

Verified step by step guidance

1. Step 1: Begin by using the properties of logarithms to simplify the function. Recall that \( \ln(a \cdot b) = \ln(a) + \ln(b) \). Apply this to \( f(x) = \ln(\sec^4x \cdot \tan^2x) \) to get \( f(x) = \ln(\sec^4x) + \ln(\tan^2x) \).
2. Step 2: Further simplify using the property \( \ln(a^b) = b \cdot \ln(a) \). Apply this to each term: \( \ln(\sec^4x) = 4 \cdot \ln(\sec x) \) and \( \ln(\tan^2x) = 2 \cdot \ln(\tan x) \). Thus, \( f(x) = 4 \cdot \ln(\sec x) + 2 \cdot \ln(\tan x) \).
3. Step 3: Differentiate each term separately. For \( 4 \cdot \ln(\sec x) \), use the chain rule: \( \frac{d}{dx}[\ln(\sec x)] = \frac{1}{\sec x} \cdot \frac{d}{dx}[\sec x] \). Recall that \( \frac{d}{dx}[\sec x] = \sec x \cdot \tan x \). Therefore, \( \frac{d}{dx}[\ln(\sec x)] = \tan x \). Multiply by 4 to get \( 4 \cdot \tan x \).
4. Step 4: Differentiate the second term \( 2 \cdot \ln(\tan x) \) using the chain rule: \( \frac{d}{dx}[\ln(\tan x)] = \frac{1}{\tan x} \cdot \frac{d}{dx}[\tan x] \). Recall that \( \frac{d}{dx}[\tan x] = \sec^2 x \). Therefore, \( \frac{d}{dx}[\ln(\tan x)] = \sec^2 x \cdot \frac{1}{\tan x} = \frac{\sec^2 x}{\tan x} \). Multiply by 2 to get \( 2 \cdot \frac{\sec^2 x}{\tan x} \).
5. Step 5: Combine the derivatives from steps 3 and 4 to find \( f'(x) \). The derivative of \( f(x) = 4 \cdot \ln(\sec x) + 2 \cdot \ln(\tan x) \) is \( f'(x) = 4 \cdot \tan x + 2 \cdot \frac{\sec^2 x}{\tan x} \).

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus, representing the slope of the tangent line to the function's graph at any given point. The derivative is denoted as f'(x) and can be calculated using various rules, such as the power rule, product rule, and chain rule.

Logarithmic Properties

Logarithmic properties are rules that simplify the manipulation of logarithmic expressions. Key properties include the product rule (log(a*b) = log(a) + log(b)), the quotient rule (log(a/b) = log(a) - log(b)), and the power rule (log(a^b) = b*log(a)). These properties are particularly useful in calculus for simplifying complex functions before differentiation.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative can be found by multiplying the derivative of the outer function f with the derivative of the inner function g. This rule is essential when dealing with functions that involve nested expressions, such as logarithmic or trigonometric functions.