Master the Derivative of Sec x: Quotient Rule or Simplifying with Sine and Cosine

Dx {sec x}=?

To find the derivative of sec x, we can use the quotient rule or rewrite it in terms of sine and cosine

To find the derivative of sec x, we can use the quotient rule or rewrite it in terms of sine and cosine.

Using Quotient Rule:
The quotient rule states that if we have a function f(x) = g(x) / h(x), then its derivative is given by:

f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / [h(x)]^2

Applying the quotient rule to the function f(x) = 1 / cos x, we have:

f'(x) = [(0 * cos x) – (1 * (-sin x))] / [cos x]^2
= sin x / cos^2 x
= sin x / cos x * cos x
= sin x * sec x

Therefore, the derivative of sec x, denoted as Dx {sec x}, is equal to sin x * sec x.

Alternatively, we can rewrite sec x in terms of sine and cosine:
sec x = 1 / cos x
= cos x / cos^2 x
= cos x * sec x

Therefore, the derivative of sec x, Dx {sec x}, is equal to cos x * sec x.

Both sin x * sec x and cos x * sec x can represent the derivative of sec x.

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