In Exercises 59–86, find the derivative of y with respect to the given independent variable.
61. y = 5√s

Verified step by step guidance

1. Identify the given function: \( y = 5\sqrt{s} \). Recall that \( \sqrt{s} \) can be rewritten as \( s^{\frac{1}{2}} \).
2. Rewrite the function using exponent notation: \( y = 5s^{\frac{1}{2}} \). This makes it easier to apply the power rule for differentiation.
3. Apply the constant multiple rule: The derivative of \( 5s^{\frac{1}{2}} \) is \( 5 \) times the derivative of \( s^{\frac{1}{2}} \).
4. Use the power rule for derivatives: For \( s^{n} \), the derivative with respect to \( s \) is \( n s^{n-1} \). So, \( \frac{d}{ds} s^{\frac{1}{2}} = \frac{1}{2} s^{-\frac{1}{2}} \).
5. Combine the results: Multiply the constant 5 by the derivative from the power rule to get \( \frac{dy}{ds} = 5 \times \frac{1}{2} s^{-\frac{1}{2}} \).

Key Concepts

Derivative of a Power Function

The derivative of a function in the form y = s^n is found using the power rule, which states that dy/ds = n * s^(n-1). This rule simplifies differentiation of roots and powers by expressing roots as fractional exponents.

Rewriting Roots as Exponents

A root such as the square root of s can be rewritten as s^(1/2). This conversion allows the use of the power rule for differentiation, making it easier to handle roots in calculus problems.

Constant Multiple Rule

When differentiating a function multiplied by a constant, the constant can be factored out and multiplied by the derivative of the variable part. For example, if y = 5 * s^(1/2), then dy/ds = 5 * d/ds(s^(1/2)).