Derivative of Tan 2x

We now understand the derivative of tan 2x.

It is commonly represented as d/dx (tan 2x) or (tan 2x)', and its value is 2 sec²(2x).

The function tan 2x has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Tangent Function: tan(2x) = sin(2x)/cos(2x).

Quotient Rule: Rule for differentiating tan(2x) (since it consists of sin(2x)/cos(2x)).

Secant Function: sec(x) = 1/cos(x).

The derivative of tan 2x can be denoted as d/dx (tan 2x) or (tan 2x)'. The formula we use to differentiate tan 2x is: d/dx (tan 2x) = 2 sec²(2x) (or) (tan 2x)' = 2 sec²(2x) The formula applies to all x where cos(2x) ≠ 0.

We can derive the derivative of tan 2x 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 tan 2x results in 2 sec²(2x) using the above-mentioned methods:

By First Principle

The derivative of tan 2x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of tan 2x using the first principle, we will consider f(x) = tan 2x.

Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1)

Given that f(x) = tan 2x, we write f(x + h) = tan (2(x + h)).

Substituting these into equation (1), f'(x) = limₕ→₀ [tan(2(x + h)) - tan 2x] / h = limₕ→₀ [ [sin (2(x + h)) / cos (2(x + h))] - [sin 2x / cos 2x] ] / h = limₕ→₀ [ [sin (2x + 2h) cos 2x - cos (2x + 2h) sin 2x] / [cos 2x · cos(2x + 2h)] ]/ h

We now use the formula sin A cos B - cos A sin B = sin (A - B). f'(x) = limₕ→₀ [ sin (2h) ] / [ h cos 2x · cos(2x + 2h)] = limₕ→₀ [ sin 2h ] / [ h cos 2x · cos(2x + 2h)] = limₕ→₀ 2(sin h)/ h · limₕ→₀ 1 / [cos 2x · cos(2x + 0)]

Using limit formulas, limₕ→₀ (sin h)/ h = 1. f'(x) = 2 [ 1 / (cos 2x · cos(2x))] = 2/cos²(2x) As the reciprocal of cosine is secant, we have, f'(x) = 2 sec²(2x).

Hence, proved.

Using Chain Rule

To prove the differentiation of tan 2x using the chain rule, We use the formula:

Tan 2x = sin 2x / cos 2x Consider f(x) = sin 2x and g(x) = cos 2x So we get, tan 2x = 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 2x and g(x) = cos 2x in equation (1), d/dx (tan 2x) = [(2 cos 2x) (cos 2x) - (sin 2x) (-2 sin 2x)] / (cos 2x)² (2 cos²(2x) + 2 sin²(2x)) / cos²(2x) …(2)

Here, we use the formula: (cos²(2x)) + (sin²(2x)) = 1 (Pythagorean identity)

Substituting this into (2), d/dx (tan 2x) = 2 / (cos²(2x)) Since sec x = 1/cos x, we write: d/dx(tan 2x) = 2 sec²(2x)

Using Product Rule

We will now prove the derivative of tan 2x using the product rule.

The step-by-step process is demonstrated below:

Here, we use the formula,

Tan 2x = sin 2x / cos 2x tan 2x = (sin 2x). (cos 2x)⁻¹

Given that, u = sin 2x and v = (cos 2x)⁻¹

Using the product rule formula: d/dx [u.v] = u'. v + u. v' u' = d/dx (sin 2x) = 2 cos 2x. (substitute u = sin 2x)

Here we use the chain rule: v = (cos 2x)⁻¹ = (cos 2x)⁻¹ (substitute v = (cos 2x)⁻¹) v' = -1. (cos 2x)⁻². d/dx (cos 2x) v' = 2 sin 2x / (cos 2x)²

Again, use the product rule formula: d/dx (tan 2x) = u'. v + u. v'

Let's substitute u = sin 2x, u' = 2 cos 2x, v = (cos 2x)⁻¹, and v' = 2 sin 2x / (cos 2x)²

When we simplify each term: We get, d/dx (tan 2x) = 2 + 2 sin²(2x) / (cos 2x)² Sin²(2x) / (cos 2x)² = tan²(2x) (we use the identity sin²(2x) + cos²(2x) = 1) Thus: d/dx (tan 2x) = 2 + 2 tan²(2x)

Since, 1 + tan²(2x) = sec²(2x) d/dx (tan 2x) = 2 sec²(2x).

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 tan(2x). 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 tan(2x), 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/4, the derivative is 2 sec²(π/2), which is undefined because tan(2x) has a vertical asymptote there. When x is 0, the derivative of tan 2x = 2 sec²(0), which is 2.

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

• Tangent Function: The tangent function is one of the primary six trigonometric functions and is written as tan x.

• Secant Function: A trigonometric function that is the reciprocal of the cosine function. It is typically represented as sec x.

• Chain Rule: A rule for finding the derivative of the composition of two functions.

• First Principle: A method of deriving the derivative of a function by using the concept of limits.