Derivative of 5sinx

We now understand the derivative of 5sin(x). It is commonly represented as d/dx (5sinx) or (5sinx)', and its value is 5cos(x).
 

The function 5sin(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 trigonometric function.

- Constant Multiple Rule: Rule for differentiating 5sin(x) (since it involves a constant multiple of sin(x)).

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

The derivative of 5sin(x) can be denoted as d/dx (5sinx) or (5sinx)'.
 

The formula we use to differentiate 5sin(x) is: d/dx (5sinx) = 5cos(x) (5sinx)' = 5cos(x) The formula applies to all x.

We can derive the derivative of 5sin(x) using proofs. To show this, we will use trigonometric identities along with the rules of differentiation. There are several methods we use to prove this, such as:

- By First Principle - Using Constant Multiple Rule We will now demonstrate that the differentiation of 5sin(x) results in 5cos(x) using the above-mentioned methods:

By First Principle The derivative of 5sin(x) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of 5sin(x) using the first principle, we will consider f(x) = 5sin(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) = 5sin(x), we write f(x + h) = 5sin(x + h). Substituting these into equation (1), f'(x) = limₕ→₀ [5sin(x + h) - 5sin(x)] / h = 5 limₕ→₀ [sin(x + h) - sin(x)] / h We use the formula sin A - sin B = 2cos((A+B)/2)sin((A-B)/2). = 5 limₕ→₀ [2cos((2x + h)/2)sin(h/2)] / h = 5 limₕ→₀ [cos((2x + h)/2)] . limₕ→₀ [sin(h/2)/(h/2)]

Using limit formulas, limₕ→₀ [sin(h)/h] = 1. f'(x) = 5cos(x) Hence, proved.

Using Constant Multiple Rule To prove the differentiation of 5sin(x) using the constant multiple rule, We use the formula: d/dx [c·f(x)] = c·d/dx [f(x)] Let c = 5 and f(x) = sin(x) d/dx (5sinx) = 5·d/dx (sinx) = 5cos(x)

Thus, the derivative is 5cos(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 5sin(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 5sin(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 π/2, the derivative is 5cos(π/2), which is 0 because cos(π/2) = 0. When x is 0, the derivative of 5sin(x) = 5cos(0), which is 5 because cos(0) = 1.

• 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 that is a common basis for waveforms.

• Cosine Function: A trigonometric function derived as the derivative of sine and is used to describe oscillations. 

• Constant Multiple Rule: A rule that allows easy differentiation of functions multiplied by a constant.

• First Derivative: It is the initial result of a function's differentiation, which gives us the rate of change of a specific function.