Verifying derivative formulas Verify the following derivative formulas using the Quotient Rule. d/dx (sec x) = sec x tan x
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Step 1: Recall the definition of secant. The secant function is defined as \( \sec x = \frac{1}{\cos x} \).
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 \( v = \cos x \).
Step 3: Differentiate \( u \) and \( v \). Since \( u = 1 \), \( u' = 0 \). For \( v = \cos x \), \( v' = -\sin x \).
Step 4: Substitute into the Quotient Rule formula. Plug \( u, v, u', \) and \( v' \) into the formula: \( \frac{\cos x \cdot 0 - 1 \cdot (-\sin x)}{(\cos x)^2} \).
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.
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