5-8. Use differentiation to verify each equation.


d/dx (tan³ x-3 tan x+3x) = 3 tan⁴x

Verified step by step guidance

1. Step 1: Identify the function to differentiate. The function given is \( f(x) = \tan^3 x - 3 \tan x + 3x \).
2. Step 2: Differentiate each term of the function separately. Start with \( \tan^3 x \). Use the chain rule: \( \frac{d}{dx}(\tan^3 x) = 3 \tan^2 x \cdot \sec^2 x \).
3. Step 3: Differentiate the second term \( -3 \tan x \). The derivative of \( \tan x \) is \( \sec^2 x \), so \( \frac{d}{dx}(-3 \tan x) = -3 \sec^2 x \).
4. Step 4: Differentiate the third term \( 3x \). The derivative of \( 3x \) is simply \( 3 \).
5. Step 5: Combine the derivatives from each term to find the derivative of the entire function: \( \frac{d}{dx}(\tan^3 x - 3 \tan x + 3x) = 3 \tan^2 x \sec^2 x - 3 \sec^2 x + 3 \). Simplify and verify if it equals \( 3 \tan^4 x \).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative represents the rate of change of a function with respect to its variable. It is essential for analyzing the behavior of functions, including their slopes and rates of increase or decrease. In this context, differentiation is used to verify the correctness of the given equation by calculating the derivative of the left-hand side.

Chain Rule

The Chain Rule is a key technique in differentiation that allows us to differentiate composite functions. It states that if a function is composed of two or more functions, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. This rule is particularly useful when dealing with functions like tan³(x), where the outer function is the cube and the inner function is the tangent function.

Power Rule

The Power Rule is a basic rule in differentiation that simplifies the process of finding the derivative of polynomial functions. It states that the derivative of x^n is n*x^(n-1), where n is a constant. This rule is applicable in the given equation for differentiating terms like tan³(x) and 3x, making it easier to compute the overall derivative and verify the equation.