Derivative of 2sinx

We now understand the derivative of 2sinx.

It is commonly represented as d/dx (2sinx) or (2sinx)', and its value is 2cosx.

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

The key concepts are mentioned below:

Sine Function: sin(x) is a basic trigonometric function.

Derivative of Sine: The derivative of sin(x) is cos(x).

Constant Multiplication Rule: If a function is multiplied by a constant, its derivative is the constant multiplied by the derivative of the function.

The derivative of 2sinx can be denoted as d/dx (2sinx) or (2sinx)'. The formula we use to differentiate 2sinx is: d/dx (2sinx) = 2cosx (or) (2sinx)' = 2cosx The formula applies to all x.

We can derive the derivative of 2sinx 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 the Constant Multiple Rule

Using Chain Rule

We will now demonstrate that the differentiation of 2sinx results in 2cosx using the above-mentioned methods:

By First Principle

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

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

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

Given that f(x) = 2sinx, we write f(x + h) = 2sin(x + h). Substituting these into equation (1), f'(x) = limₕ→₀ [2sin(x + h) - 2sinx] / h = limₕ→₀ 2[sin(x + h) - sinx] / h = 2 limₕ→₀ [sin(x + h) - sinx] / h

Using the formula for the derivative of sinx, = 2cosx

Hence, proved.

Using Constant Multiplication Rule

To prove the differentiation of 2sinx using the constant multiplication rule,

We use the formula: d/dx [c·f(x)] = c·f'(x) Let c = 2 and f(x) = sinx, so we have, d/dx (2sinx) = 2·d/dx (sinx) = 2cosx

Using Chain Rule

To prove the differentiation of 2sinx using the chain rule, Consider y = 2sinx = 2·sinx

Using the chain rule, dy/dx = 2 · d/dx (sinx) = 2cosx

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 2sin(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 2sin(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 0 or any multiple of π, the derivative 2cos(x) is not zero since cos(0) = 1, so the derivative is 2. The function 2sinx is continuous and differentiable for all x.

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

• Sine Function: A trigonometric function represented as sin(x).

• Cosine Function: A trigonometric function that is the derivative of sin(x) and is represented as cos(x).

• Constant Multiplication Rule: A rule stating that the derivative of a constant times a function is the constant times the derivative of the function.

• Chain Rule: A rule used to differentiate composite functions.