Verifying derivative formulas Verify the following derivative formulas using the Quotient Rule.
d/dx (sec x) = sec x tan x

Verified step by step guidance

1. Step 1: Recall the definition of secant. The secant function is defined as \( \sec x = \frac{1}{\cos x} \).
2. Step 2: Apply the Quotient Rule. The Quotient Rule states that if you have a function \( \frac{u}{v} \), its derivative is \( \frac{v \cdot u' - u \cdot v'}{v^2} \). Here, \( u = 1 \) and \( v = \cos x \).
3. Step 3: Differentiate \( u \) and \( v \). Since \( u = 1 \), \( u' = 0 \). For \( v = \cos x \), \( v' = -\sin x \).
4. Step 4: Substitute into the Quotient Rule formula. Plug \( u, v, u', \) and \( v' \) into the formula: \( \frac{\cos x \cdot 0 - 1 \cdot (-\sin x)}{(\cos x)^2} \).
5. Step 5: Simplify the expression. The expression simplifies to \( \frac{\sin x}{\cos^2 x} \), which can be rewritten as \( \sec x \tan x \) using trigonometric identities.

Key Concepts

Quotient Rule

The Quotient Rule is a fundamental technique in calculus used to differentiate functions that are expressed as the ratio of two other functions. If you have a function f(x) = g(x)/h(x), the derivative f'(x) is given by (g'(x)h(x) - g(x)h'(x)) / (h(x))^2. This rule is essential for verifying derivatives of functions that are not simple polynomials.

Trigonometric Derivatives

Trigonometric derivatives refer to the derivatives of trigonometric functions, which are foundational in calculus. For example, the derivative of sec(x) is derived from the basic derivatives of sine and cosine functions. Understanding these derivatives is crucial for applying rules like the Quotient Rule effectively, especially when dealing with sec(x) and its relationship to cosine.

Chain Rule

The Chain Rule is another important differentiation technique used when dealing with composite functions. It states that if you have a function h(x) = f(g(x)), then the derivative h'(x) is f'(g(x)) * g'(x). While the Quotient Rule is used for ratios, the Chain Rule often comes into play when differentiating functions like sec(x), which can be expressed in terms of other trigonometric functions.