Derivative of Sin Cubed x

We now understand the derivative of sin³(x). It is commonly represented as d/dx (sin³x) or (sin³x)', and its value is 3sin²x·cosx. The function sin³x has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below:

Sine Function: (sin(x)).

Chain Rule: Rule for differentiating composite functions like sin³x.

Power Rule: Rule for differentiating functions of the form x^n.

The derivative of sin³x can be denoted as d/dx (sin³x) or (sin³x)'.

The formula we use to differentiate sin³x is: d/dx (sin³x) = 3sin²x·cosx

The formula applies to all x where sin(x) is defined.

We can derive the derivative of sin³x using proofs. To show this, we will use trigonometric identities along with differentiation rules. There are several methods we use to prove this, such as: Using the Chain Rule Using the Product Rule We will now demonstrate that the differentiation of sin³x results in 3sin²x·cosx using these methods:

Using the Chain Rule

To prove the differentiation of sin³x using the chain rule, We use the formula: Let y = (sin(x))³

By the chain rule, dy/dx = 3(sin(x))²·d/dx(sin(x))

Since d/dx(sin(x)) = cos(x), dy/dx = 3(sin(x))²·cos(x) Hence, proved.

Using the Product Rule

We can also prove the derivative of sin³x using the product rule. Here's the step-by-step process: Let y = sin(x)·sin(x)·sin(x)

By the product rule, d/dx[u·v·w] = uv'w + u'vw + uvw' Let u = sin(x), v = sin(x), w = sin(x)

Then u' = cos(x), v' = cos(x), w' = cos(x) dy/dx = sin(x)·sin(x)·cos(x) + sin(x)·cos(x)·sin(x) + cos(x)·sin(x)·sin(x) = 3sin²(x)·cos(x)

Thus, d/dx(sin³x) = 3sin²x·cosx.

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little complex.

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 help in understanding functions like sin³(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 sin³(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.

When x is 0, the derivative of sin³(x) = 3sin²(0)·cos(0), which is 0.

• 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 and is written as sin(x).

• Chain Rule: A rule for differentiating composite functions, where you differentiate the outer function and multiply by the derivative of the inner function.

• Power Rule: A rule for differentiating functions of the form x^n, which states that the derivative is n*x^(n-1).

• Quotient Rule: A rule for differentiating the quotient of two functions, where the derivative is given by (v·u' - u·v')/v².