Derivative of Sec θ

We now understand the derivative of sec θ.

It is commonly represented as d/dθ (sec θ) or (sec θ)', and its value is sec θ tan θ.

The function sec θ has a well-defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Secant Function: sec(θ) = 1/cos(θ).

Tangent Function: tan(θ) = sin(θ)/cos(θ).

Product Rule: Rule for differentiating sec(θ) using its product form.

The derivative of sec θ can be denoted as d/dθ (sec θ) or (sec θ)'. The formula we use to differentiate sec θ is: d/dθ (sec θ) = sec θ tan θ The formula applies to all θ where cos(θ) ≠ 0.

We can derive the derivative of sec θ using proofs.

To show this, we will use trigonometric identities along with differentiation rules.

There are several methods for proving this, such as:

By First Principle

Using Chain Rule

Using Product Rule

We will now demonstrate that the differentiation of sec θ results in sec θ tan θ using these methods:

By First Principle

The derivative of sec θ can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of sec θ using the first principle, we will consider f(θ) = sec θ. Its derivative can be expressed as the following limit.

f'(θ) = limₕ→₀ [f(θ + h) - f(θ)] / h … (1)

Given that f(θ) = sec θ, we write f(θ + h) = sec(θ + h).

Substituting these into equation (1), f'(θ) = limₕ→₀ [sec(θ + h) - sec θ] / h = limₕ→₀ [1/cos(θ + h) - 1/cos θ] / h = limₕ→₀ [cos θ - cos(θ + h)] / [h cos(θ) cos(θ + h)]

Using the identity cos A - cos B = -2 sin((A + B)/2) sin((A - B)/2), f'(θ) = limₕ→₀ [-2 sin((2θ + h)/2) sin(h/2)] / [h cos(θ) cos(θ + h)] = limₕ→₀ [-2 sin(θ + h/2) sin(h/2) / h] 1/[cos(θ) cos(θ + h)]

Using limit formulas, limₕ→₀ (sin h/2)/(h/2) = 1. f'(θ) = [sin θ / cos² θ] = sec θ tan θ.

Hence, proved.

Using Chain Rule

To prove the differentiation of sec θ using the chain rule, We use the formula: sec θ = 1/cos θ Let f(θ) = 1/u, where u = cos θ

Using the chain rule: d/dθ [1/u] = -1/u² · du/dθ

Let’s substitute u = cos θ in the equation, d/dθ (sec θ) = -1/(cos θ)² · (-sin θ) = sin θ/(cos θ)² Since sec θ = 1/cos θ, we write: d/dθ(sec θ) = sec θ tan θ

Using Product Rule

We will now prove the derivative of sec θ using the product rule.

The step-by-step process is demonstrated below:

Here, we use the formula, sec θ = (cos θ)⁻¹ Given that, u = 1 and v = (cos θ)⁻¹

Using the product rule formula: d/dθ [u.v] = u'v + uv' u' = d/dθ (1) = 0 v' = d/dθ ((cos θ)⁻¹) = sin θ/(cos θ)²

Again, using the product rule formula: d/dθ (sec θ) = 0 · v + 1 · v'

Substitute u = 1, u' = 0, v = (cos θ)⁻¹, and v' = sin θ/(cos θ)² d/dθ (sec θ) = sin θ/(cos θ)²

Thus: d/dθ (sec θ) = sec θ tan θ.

When a function is differentiated multiple times, the results are known as higher-order derivatives. Higher-order derivatives can be a bit complex.

To understand them better, consider a car where the speed changes (first derivative) and the rate of change of speed (second derivative) also changes.

Higher-order derivatives help in understanding functions like sec(θ).

For the first derivative of a function, we write f′(θ), which indicates how the function changes or its slope at a certain point.

The second derivative is derived from the first derivative, denoted as f′′(θ).

Similarly, the third derivative, f′′′(θ), is the result of the second derivative, and this pattern continues.

For the nth Derivative of sec(θ), we generally use fⁿ(θ) for the nth derivative of a function f(θ), which tells us the change in the rate of change (continuing for higher-order derivatives).

When θ is π/2 or any odd multiple of π/2, the derivative is undefined because sec(θ) has a vertical asymptote there. When θ is 0, the derivative of sec θ = sec(0) tan(0), which is 0.

• Derivative: The derivative of a function indicates how the given function changes in response to a slight change in θ.

• Secant Function: A trigonometric function that is the reciprocal of the cosine function, represented as sec θ.

• Tangent Function: The tangent function, written as tan θ, is one of the primary trigonometric functions.

• Product Rule: A rule used to find the derivative of the product of two functions.

• Asymptote: A line that a graph approaches but never touches or crosses.