Derivative of sec(x): Proof and Examples

Step-by-Step Quotient Rule Derivation, Graph, Chain Rule Formula, and Solved Examples

The derivative of \( \sec(x) \) is calculated using the quotient rule of derivatives.

Proof of the Derivative of sec(x)

A trigonometric identity relating \( \sec x \) and \( \cos x \) is given by:

\[ \sec x = \dfrac{1}{\cos x} \]

We use the quotient rule of differentiation to find the derivative of \( \sec x \):

\[ \dfrac{d}{dx} \sec x = \dfrac{d}{dx} \left(\dfrac{1}{\cos x}\right) = \dfrac{\left(\dfrac{d}{dx}1\right) \cos x - 1 \left(\dfrac{d}{dx} \cos x\right)}{\cos^2 x} \]

The derivative of the constant 1 is equal to zero. Using the formula for the derivative of the trigonometric function \( \cos x \), which is \( \dfrac{d}{dx}\cos x = -\sin x \), substitute to obtain:

\[ \dfrac{d}{dx} \sec x = \dfrac{0 - (-\sin x)}{\cos^2 x} \]

Simplify:

\[ = \dfrac{\sin x}{\cos^2 x} = \dfrac{\sin x}{\cos x} \cdot \dfrac{1}{\cos x} = \tan x \sec x \]
Formula: \[ \dfrac{d}{dx} \sec x = \tan x \; \sec x \]

Graph of sec(x) and Its Derivative

The graphs of \( \sec(x) \) and its derivative are shown below:

Graph of sec x and its derivative

Derivative of the Composite Function sec(u(x))

We now consider the composite function secant of another function \( u(x) \). Use the chain rule of differentiation to write:

\[ \dfrac{d}{dx} \sec(u(x)) = \left(\dfrac{d}{du} \sec u\right) \left(\dfrac{d}{dx} u\right) \] \[ = \tan u \sec u \dfrac{du}{dx} \]
Chain Rule Formula: \[ \dfrac{d}{dx} \sec(u(x)) = \tan(u(x)) \; \sec(u(x)) \; u'(x) \]

Examples with Solutions

Worked Examples: Differentiating Composite Secant Functions

Find the derivative of the composite secant functions:

  1. \( f(x) = \sec(x^2 + x - 1) \)
  2. \( g(x) = \sec(\sin x) \)
  3. \( h(x) = \sec(\sqrt{x+2}) \)

Solutions:

  1. Let \( u(x) = x^2 + x - 1 \), then \( \dfrac{du}{dx} = 2x + 1 \).
    Applying the rule for the composite secant function: \[ f'(x) = \tan(u) \sec(u) \dfrac{du}{dx} = (2x + 1) \, \tan(x^2 + x - 1) \, \sec(x^2 + x - 1) \]
  2. Let \( u(x) = \sin x \), then \( \dfrac{du}{dx} = \cos x \).
    Applying the rule: \[ g'(x) = \tan(u) \sec(u) \dfrac{du}{dx} = \cos x \, \tan(\sin x) \, \sec(\sin x) \]
  3. Let \( u(x) = \sqrt{x+2} \), then \( \dfrac{du}{dx} = \dfrac{1}{2\sqrt{x+2}} \).
    Applying the rule: \[ h'(x) = \tan(u) \sec(u) \dfrac{du}{dx} = \dfrac{\tan(\sqrt{x+2}) \, \sec(\sqrt{x+2})}{2\sqrt{x+2}} \]

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