91Ó°ÊÓ

The constant multiple rule is a fundamental concept in differentiation. It helps us find the derivative of a function that is multiplied by a constant. When you have a function of the form \( kf(x) \), where \( k \) is a constant and \( f(x) \) is a function of \( x \), the constant multiple rule states that the derivative of \( kf(x) \) with respect to \( x \) is \( k \) times the derivative of \( f(x) \). This simplifies our process significantly.

For example, in the given exercise, we have \( y = 10 \times \text{sin}(x) \). Here, \( 10 \) is the constant and \( \text{sin}(x) \) is our function \( f(x) \). To apply the constant multiple rule, we first recognize the form and then multiply the constant \( 10 \) by the derivative of \( \text{sin}(x) \). This concept is very powerful and can be applied to any function that is multiplied by a constant in calculus.

Trigonometric functions like sine, cosine, and tangent are essential in calculus. In the given problem, we deal with the sine function, denoted as \( \text{sin}(x) \). Understanding the derivatives of basic trigonometric functions is crucial for solving more complex problems.

The derivative of \( \text{sin}(x) \) with respect to \( x \) is \( \text{cos}(x) \). This relationship is important because it allows us to apply the constant multiple rule effectively. In other words, when differentiating \( 10 \text{sin}(x) \), knowing that \( \frac{d}{dx} (\text{sin}(x)) = \text{cos}(x) \) helps us quickly find the solution.

Always remember these basic derivatives for common trigonometric functions:

• \( \frac{d}{dx} (\text{sin}(x)) = \text{cos}(x) \)
• \( \frac{d}{dx} (\text{cos}(x)) = -\text{sin}(x) \)
• \( \frac{d}{dx} (\text{tan}(x)) = \text{sec}^2(x) \)

Knowing these derivatives will save you time and make solving trigonometric function problems easier.

Calculating derivatives is the cornerstone of differentiation in calculus. The process involves finding how a function changes at any given point. Here, we specifically calculate the derivative of \( 10 \text{sin}(x) \) step by step.

First, recognize that \( 10 \text{sin}(x) \) is a product of the constant \( 10 \) and the sine function. Using the constant multiple rule, we separate the constant from the function, giving us \( 10 \frac{d}{dx} (\text{sin}(x)) \).

Next, differentiate \( \text{sin}(x) \) to get \( \text{cos}(x) \), as we know from trigonometric derivatives. Putting this all together, we multiply \( 10 \) by \( \text{cos}(x) \), resulting in \( 10 \text{cos}(x) \).

To summarize, the steps are:

• Identify the function form.
• Apply the constant multiple rule by separating the constant.
• Differentiate the trigonometric function.
• Combine the results to get the final derivative.

By carefully following these steps, we ensure accurate and efficient derivative calculations.