Derivative of Sec x

We now understand the derivative of sec x. It is commonly represented as d/dx (sec x) or (sec x)', and its value is sec(x)tan(x). The function sec x has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Secant Function: sec(x) = 1/cos(x). Quotient Rule: Rule for differentiating sec(x) since it consists of 1/cos(x). Tangent Function: tan(x) = sin(x)/cos(x).

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

We can derive the derivative of sec x using proofs. To show this, we will use the trigonometric identities along with the rules of differentiation. There are several methods we use to prove this, such as: By First Principle Using Chain Rule Using Product Rule We will now demonstrate that the differentiation of sec x results in sec(x)tan(x) using the above-mentioned methods: By First Principle The derivative of sec x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of sec x using the first principle, we will consider f(x) = sec x. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = sec x, we write f(x + h) = sec(x + h). Substituting these into equation (1), f'(x) = limₕ→₀ [sec(x + h) - sec x] / h = limₕ→₀ [(1/cos(x + h)) - (1/cos x)] / h = limₕ→₀ [(cos x - cos(x + h)) / (cos x · cos(x + h))] / h We now use the formula cos A - cos B = -2 sin((A+B)/2)sin((A-B)/2). f'(x) = limₕ→₀ [-2 sin((2x + h)/2) sin(h/2)] / [h cos x · cos(x + h)] = limₕ→₀ [-2 sin((2x + h)/2) (sin(h/2)/h)] / [cos x · cos(x + h)] Using limit formulas, limₕ→₀ (sin(h/2)/h) = 1/2. f'(x) = - sin(2x)/cos^2x As the product of secant and tangent is sec(x)tan(x), we have, f'(x) = sec(x)tan(x). Hence, proved. Using Chain Rule To prove the differentiation of sec x using the chain rule, We use the formula: Sec x = 1/cos x Consider f(x) = 1 and g (x) = cos x Then, sec x = f (x)/ g(x) By quotient rule: d/dx [f(x) / g(x)] = [f '(x) g(x) - f(x) g'(x)] / [g(x)]²… (1) Let’s substitute f(x) = 1 and g (x) = cos x in equation (1), d/ dx (sec x) = [(0) (cos x)- (1) (- sin x)]/ (cos x)² = sin x / cos² x …(2) Here, we use the formula: sin x / cos² x = tan(x) / cos(x) Substituting this into (2), d/dx (sec x) = sec(x)tan(x). Using Product Rule We will now prove the derivative of sec x using the product rule. The step-by-step process is demonstrated below: Here, we use the formula, Sec x = 1/cos x sec x = (cos x)-1 Given that, u = 1 and v = (cos x)-1 Using the product rule formula: d/dx [u.v] = u'. v + u. v' u' = d/dx (1) = 0. (substitute u = 1) Here we use the chain rule: v = (cos x)-1 = (cos x)-1 (substitute v = (cos x)-1) v' = -1. (cos x)-2. d/dx (cos x) v' = sin x / (cos x)² Again, use the product rule formula: d/dx (sec x) = u'. v + u. v' Let’s substitute u = 1, u' = 0, v = (cos x)-1, and v' = sin x / (cos x)² When we simplify each term: We get, d/dx (sec x) = sin x / (cos x)² = tan(x) / cos(x) Thus: d/dx (sec x) = sec(x)tan(x).

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little tricky. To understand them better, think of a car where the speed changes (first derivative) and the rate at which the speed changes (second derivative) also changes. Higher-order derivatives make it easier to understand functions like sec(x). For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point. The second derivative is derived from the first derivative, which is denoted using f′′(x) Similarly, the third derivative, f′′′(x) is the result of the second derivative and this pattern continues. For the nth Derivative of sec(x), we generally use fⁿ(x) for the nth derivative of a function f(x) which tells us the change in the rate of change. (continuing for higher-order derivatives).

When the x is π/2, the derivative is undefined because sec(x) has a vertical asymptote there. When the x is 0, the derivative of sec x = 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 x. Secant Function: A trigonometric function that is the reciprocal of the cosine function. It is typically represented as sec x. Tangent Function: One of the primary six trigonometric functions, written as tan x. First Derivative: It is the initial result of a function, which gives us the rate of change of a specific function. Asymptote: A line that a curve approaches as it heads towards infinity. Sec x has vertical asymptotes at points where cos x = 0.