91Ó°ÊÓ

The power rule is a basic yet powerful tool in calculus for differentiating functions of the form \( y^n \). It simplifies the process by providing a straightforward formula: \( \frac{d}{dy}[y^n] = n y^{n-1} \). This rule is especially useful when dealing with polynomial functions, as it allows for quick computation of derivatives without complex procedures. In our exercise, the function is \( y^{-6} \), where the exponent is \(-6\). Using the power rule, the derivative becomes \(-6 y^{-7} \):

• Bring down the exponent as a coefficient.
• Reduce the exponent by one.

This results in a new term, making differentiation straightforward. Remember, the power rule applies to any real number exponent, not just positive integers.

The constant multiple rule simplifies differentiating functions that include a constant coefficient. It states that if a function is multiplied by a constant, you can differentiate by multiplying the constant with the derivative of the function itself. For instance, consider a function \( f(y) = c \cdot y^n \). Its derivative can be quickly found as \( c \cdot \frac{d}{dy}[y^n] \).
In our given example, the function is \( B(y) = c y^{-6} \). We've already determined the derivative of \( y^{-6} \) to be \(-6 y^{-7} \). By using the constant multiple rule, we multiply this result by \( c \), which gives us \( -6c y^{-7} \). This rule helps ensure we maintain accuracy while simplifying calculations.

Differentiation plays a crucial role in solving calculus problems, as it provides insights into the rate of change of functions. When faced with a calculus problem requiring differentiation, it's essential to:

• Identify the function and any constants involved.
• Apply the appropriate differentiation rules (e.g., power rule, constant multiple rule).
• Simplify the result to its most basic form.

In this exercise, we started with \( B(y) = c y^{-6} \). Each step was meticulously handled using relevant rules:
1. Applying the power rule to \( y^{-6} \) to get \(-6y^{-7}\).2. Utilizing the constant multiple rule to include \( c \).3. Combining these steps to simplify the derivative as \( B'(y) = -6c y^{-7} \).By following a structured approach, you can tackle even complex calculus problems with ease and confidence.