Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
f(s) = √s/4

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Step 1: Rewrite the function f(s) = \(\frac{\sqrt{s}\)}{4} in a form that is easier to differentiate. Recall that \(\sqrt{s}\) can be expressed as s^{1/2}. Therefore, f(s) = \(\frac{s^{1/2}\)}{4}.

Step 2: Apply the constant multiple rule of differentiation. The constant multiple rule states that the derivative of a constant times a function is the constant times the derivative of the function. Here, the constant is \(\frac{1}{4}\). So, we need to find the derivative of s^{1/2} and then multiply it by \(\frac{1}{4}\).

Step 3: Differentiate s^{1/2} using the power rule. The power rule states that if f(x) = x^n, then f'(x) = nx^{n-1}. Applying this to s^{1/2}, we get \(\frac{1}{2}\)s^{-1/2}.

Step 4: Multiply the result from Step 3 by the constant \(\frac{1}{4}\) from Step 2. This gives us \(\frac{1}{4}\) \(\times\) \(\frac{1}{2}\)s^{-1/2}.

Step 5: Simplify the expression obtained in Step 4. Combine the constants \(\frac{1}{4}\) and \(\frac{1}{2}\) to get \(\frac{1}{8}\). Therefore, the derivative of f(s) is \(\frac{1}{8}\)s^{-1/2}.

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Derivatives

A derivative represents the rate at which a function changes at any given point. It is defined as the limit of the average rate of change of the function as the interval approaches zero. Derivatives are fundamental in calculus for understanding the behavior of functions, including their slopes and rates of change.

Power Rule

The Power Rule is a basic differentiation rule used to find the derivative of functions in the form of x^n, where n is a real number. According to this rule, the derivative of x^n is n*x^(n-1). This rule simplifies the process of differentiation, especially for polynomial functions and can be applied to functions involving roots.

Chain Rule

The Chain Rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function g(x) that is composed with another function f(x), the derivative is found by multiplying the derivative of the outer function by the derivative of the inner function. This rule is essential for differentiating complex functions, including those involving square roots.

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.f(s) = √s/4