Derivative of Sin/Cos

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

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

We can derive the derivative of sin x/cos 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 Quotient Rule - Using Product Rule We will now demonstrate that the differentiation of sin x/cos x results in tan(x) sec(x) using the above-mentioned methods: By First Principle The derivative of sin x/cos x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of sin x/cos x using the first principle, we will consider f(x) = sin x/cos 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) = sin x/cos x, we write f(x + h) = sin(x + h)/cos(x + h). Substituting these into equation (1), f'(x) = limₕ→0 [sin(x + h)/cos(x + h) - sin x/cos x] / h = limₕ→0 [(sin(x + h) cos x - sin x cos(x + h)) / (cos x cos(x + h))] / h We now use the formula sin A cos B - sin B cos A = sin(A - B). f'(x) = limₕ→0 [sin h] / [h cos x cos(x + h)] = limₕ→0 [(sin h)/h] * limₕ→0 [1 / (cos x cos(x + h))] Using limit formulas, limₕ→0 (sin h)/h = 1. f'(x) = 1 * [1 / (cos² x)] = 1/cos² x As the reciprocal of cosine is secant, we have, f'(x) = sec² x. Hence, proved. Using Quotient Rule To prove the differentiation of sin x/cos x using the quotient rule, We use the formula: sin x/cos x = tan x Consider f(x) = sin x and g(x) = cos x So, we get tan 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) = sin x and g(x) = cos x in equation (1), d/dx (sin x/cos x) = [(cos x)(cos x) - (sin x)(-sin x)] / (cos x)² = (cos² x + sin² x) / cos² x ... (2) Here, we use the formula: cos² x + sin² x = 1 (Pythagorean identity) Substituting this into (2), d/dx (tan x) = 1 / cos² x Since sec x = 1/cos x, we write: d/dx(tan x) = sec² x Using Product Rule We will now prove the derivative of sin x/cos x using the product rule. The step-by-step process is demonstrated below: Here, we use the formula, sin x/cos x = (sin x)(cos x)⁻¹ Given that u = sin x and v = (cos x)⁻¹ Using the product rule formula: d/dx [u.v] = u'.v + u.v' u' = d/dx (sin x) = cos x (substitute u = sin x) Here we use the chain rule: v = (cos x)⁻¹ (substitute v = (cos x)⁻¹) v' = -1 (cos x)⁻² d/dx (cos x) v' = sin x / cos² x Again, using the product rule formula: d/dx (sin x/cos x) = u'.v + u.v' Let’s substitute u = sin x, u' = cos x, v = (cos x)⁻¹, and v' = sin x / cos² x When we simplify each term: We get, d/dx (sin x/cos x) = 1 + sin²x / cos² x sin² x / cos² x = tan² x (we use the identity sin² x + cos² x = 1) Thus: d/dx (sin x/cos x) = 1 + tan² x Since 1 + tan² x = sec² x d/dx (sin x/cos x) = sec² 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 sin(x)/cos(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 sin(x)/cos(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 x is π/2, the derivative is undefined because sin(x)/cos(x) has a vertical asymptote there. When x is 0, the derivative of sin(x)/cos(x) = sec²(0), which is 1.

- Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. - Sine Function: One of the primary trigonometric functions, denoted as sin x. - Cosine Function: Another primary trigonometric function, denoted as cos x. - Tangent Function: The tangent function is derived from sine and cosine and is written as tan x = sin x/cos x. - Quotient Rule: A rule used to differentiate functions that are expressed as a quotient of two other functions.