91Ó°ÊÓ

The power rule is a fundamental principle in differentiation used to find the derivative of functions that are powers of a variable. In simple terms, it helps us calculate how a function changes as the input variable changes, when that function is expressed in the form of a power. The general formula for the power rule is:

• \( \frac{d}{dx} x^n = nx^{n-1} \) where \( n \) is any real number.

In the given problem, we have the function \( f(x) = 3 (x + 1)^{1/2} \). The term \( (x + 1)^{1/2} \) represents a power of \( x + 1 \). The power here is \( 1/2 \). By applying the power rule, we get the derivative of \( (x + 1)^{1/2} \) as \( \frac{1}{2} (x + 1)^{-1/2} \). This step involves:

• Bringing down the exponent (\( 1/2 \)) as a coefficient.
• Decreasing the original exponent by one to get \( (x + 1)^{-1/2} \).

This technique simplifies the process of differentiation significantly, turning it into an easy-to-apply rule for any polynomial or root-based expression.

The chain rule is another essential concept in calculus, enabling the differentiation of composite functions. A composite function is essentially a function within a function, and the chain rule helps us find how the outer function changes with respect to the inner function and ultimately the variable. The chain rule is expressed as:

• \( \frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x) \)

In our problem, we are differentiating \( f(x) = 3 (x + 1)^{1/2} \). Here, \( (x + 1) \) serves as the inside function \( g(x) \) with \( g(x) = x + 1 \) and \( g'(x) = 1 \). The outside function \( f(g(x)) = (x + 1)^{1/2} \) is differentiated as per the power rule to get \( \frac{1}{2} (x + 1)^{-1/2} \). Applying the chain rule means:

• Multiplying the derivative of the outside function by the derivative of the inside function.
• In this case, we multiply by \( g'(x) \), which is \( 1 \), keeping the derivative unchanged.

This method is crucial for dealing with more complex functions that include nested components, allowing for a seamless process of differentiation.

A derivative in calculus represents the rate at which a function is changing at any given point. It provides a way to calculate the instantaneous slope of the function’s graph at a particular point. For a function \( f(x) \), the derivative is denoted as \( f'(x) \) or \( \frac{df}{dx} \). This concept is the cornerstone of differential calculus, enabling us to understand how changes in input affect the output of a function.
The specific problem asks us to differentiate \( f(x) = 3 \sqrt{x+1} \). By rewriting it as \( f(x) = 3 (x+1)^{1/2} \) and applying the power rule followed by the chain rule, we compute:

• The derivative of the internal expression \( (x+1)^{1/2} \) with respect to \( x \).
• Adjust it by multiplying through by the constant outside, which is 3.

Our final expression becomes \( \frac{3}{2} (x+1)^{-1/2} \), or equivalently \( \frac{3}{2 \sqrt{x+1}} \). This derivative indicates how the function \( f(x) = 3 \sqrt{x+1} \) changes for very small changes in \( x \), providing valuable insights into the function's behavior over various domains.