Derivative of Sin3x

We now understand the derivative of sin(3x). It is commonly represented as d/dx (sin(3x)) or (sin(3x))', and its value is 3cos(3x).

The function sin(3x) has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below: Sine Function: The sine function is a fundamental trigonometric function.

Chain Rule: A rule for differentiating composite functions like sin(3x).

Cosine Function: The derivative of the sine function involves the cosine function.

The derivative of sin(3x) can be denoted as d/dx (sin(3x)) or (sin(3x))'.

The formula we use to differentiate sin(3x) is: d/dx (sin(3x)) = 3cos(3x) (sin(3x))' = 3cos(3x) The formula applies to all x.

We can derive the derivative of sin(3x) 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 sin(3x) results in 3cos(3x) using the above-mentioned methods:

By First Principle The derivative of sin(3x) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of sin(3x) using the first principle, we will consider f(x) = sin(3x).

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(3x), we write f(x + h) = sin(3(x + h)).

Substituting these into equation (1), f'(x) = limₕ→₀ [sin(3(x + h)) - sin(3x)] / h = limₕ→₀ [2cos(3x + 3h/2)sin(3h/2)] / h

Using the limit formula for sin(θ)/θ = 1 as θ approaches 0, f'(x) = 3cos(3x) Hence, proved.

Using Chain Rule To prove the differentiation of sin(3x) using the chain rule, Let g(x) = 3x and f(g) = sin(g). Then, f(x) = sin(3x) = sin(g(x)).

Using the chain rule: d/dx [f(g(x))] = f'(g(x)) * g'(x) f'(x) = cos(3x) * 3 = 3cos(3x) Thus, d/dx (sin(3x)) = 3cos(3x).

Using Product Rule We will now prove the derivative of sin(3x) using the product rule.

The step-by-step process is demonstrated below: We express sin(3x) as a product: sin(3x) = sin(3·x)

Given that, u = sin(3) and v = x Using the product rule formula: d/dx [u·v] = u'v + uv' u' = 0 (since sin(3) is constant) v' = 1 d/dx (sin(3x)) = 0·x + sin(3)·1 Simplifying, using chain rule: d/dx (sin(3x)) = 3cos(3x) Thus, the derivative of sin(3x) is 3cos(3x).

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(3x).

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(3x), 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 such that sin(3x) is at a maximum or minimum, the derivative is zero because the slope is horizontal at those points. When x is 0, the derivative of sin(3x) = 3cos(0) = 3.

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

• Chain Rule: A fundamental differentiation rule used to differentiate composite functions like sin(3x).

• Sine Function: A primary trigonometric function written as sin(x), with sin(3x) being a specific form.

• Cosine Function: A trigonometric function that appears in the derivative of the sine function.

• Trigonometric Function: Functions related to angles, commonly used in calculus and analysis, including sine and cosine.