Derivative of 4sinx

We now understand the derivative of 4sin x. It is commonly represented as d/dx (4sin x) or (4sin x)', and its value is 4cos x. The function 4sin x 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).

Constant Multiplication Rule: Rule for differentiating 4sin(x) (since it is a constant multiplied by a function).

Cosine Function: cos(x) is the derivative of sin(x).

The derivative of 4sin x can be denoted as d/dx (4sin x) or (4sin x)'.

The formula we use to differentiate 4sin x is: d/dx (4sin x) = 4cos x (or) (4sin x)' = 4cos x

The formula applies to all x within the domain of the sine function.

We can derive the derivative of 4sin 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:

1. By First Principle
2. Using Chain Rule
3. Using Product Rule

We will now demonstrate that the differentiation of 4sin x results in 4cos x using the above-mentioned methods:

By First Principle The derivative of 4sin x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of 4sin x using the first principle, we will consider f(x) = 4sin 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) = 4sin x, we write f(x + h) = 4sin (x + h).

Substituting these into equation (1), f'(x) = limₕ→₀ [4sin(x + h) - 4sin x] / h = 4 · limₕ→₀ [sin(x + h) - sin x] / h

Using the trigonometric identity sin(A + B) = sinAcosB + cosAsinB, f'(x) = 4 · limₕ→₀ [(sin x cos h + cos x sin h) - sin x] / h = 4 · limₕ→₀ [cos x sin h + (cos h - 1) sin x] / h = 4 · [cos x · limₕ→₀ (sin h)/ h + sin x · limₕ→₀ (cos h - 1)/ h]

Using limit formulas, limₕ→₀ (sin h)/ h = 1 and limₕ→₀ (cos h - 1)/ h = 0. f'(x) = 4 · [cos x · 1 + sin x · 0] f'(x) = 4cos x. Hence, proved.

Using Chain Rule To prove the differentiation of 4sin x using the chain rule, We use the formula: 4sin x = 4 · sin x Consider y = sin x, So we get, 4sin x = 4y By the chain rule: d/dx (4y) = 4 · d/dx (y) … (1)

Let’s substitute y = sin x in equation (1), d/dx (4sin x) = 4 · d/dx (sin x) = 4cos x

Using Product Rule We will now prove the derivative of 4sin x using the product rule. The step-by-step process is demonstrated below: Here, we use the formula, 4sin x = 4 · sin x Given that, u = 4 and v = sin x

Using the product rule formula: d/dx [u·v] = u'·v + u·v' u' = d/dx (4) = 0. (substitute u = 4) v' = d/dx (sin x) = cos x (substitute v = sin x)

Again, use the product rule formula: d/dx (4sin x) = u'·v + u·v'

Let’s substitute u = 4, u' = 0, v = sin x, and v' = cos x When we simplify each term: We get, d/dx (4sin x) = 0 · sin x + 4 · cos x Thus: d/dx (4sin x) = 4cos 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 4sin(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 4sin(x), we generally use f⁽ⁿ⁾(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 of 4sin x = 4cos(π/2), which is 0. When the x is 0, the derivative of 4sin x = 4cos(0), which is 4.

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

• Sine Function: The sine function is one of the primary trigonometric functions, represented as sin x.

• Cosine Function: A trigonometric function that is the derivative of the sine function, represented as cos x.

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

• Chain Rule: A rule for differentiating compositions of functions, used especially when differentiating products or powers of functions.