Problem 38 - Precalculus with Limits

Question

In Exercises 29-42, find the derivative of the function. \(f(x) = \sqrt{x+8}\)

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = \sqrt{x+8}\) is \(f'(x) = \frac{1}{2\sqrt{x+8}}\)

Step 1: Rewrite the function

Rewrite the function \(f(x)\) in the form that can be easily differentiated. In this case, the square root can be written as a power of one-half, so we have \(f(x) = (x+8)^{1/2}\)

Step 2: Apply the Chain Rule

The chain rule states that the derivative of a composite function is the derivative of the outer function, times the derivative of the inner function. Apply it to differentiate \(f(x) = (x+8)^{1/2}\), i.e. \(f'(x) = \frac{1}{2} (x+8)^{-1/2} \cdot (1)\)

Step 3: Simplify the Result

Multiply the result by the derivative of the inner function which is 1. Then, simplify the expression to get \(f'(x) = \frac{1}{2\sqrt{x+8}}\)

Key Concepts

Chain Rule

The chain rule is a formula to compute the derivative of a composite function. In simpler terms, when you have a function tucked inside another, the chain rule helps you find the rate at which the composite whole changes. To apply the chain rule, you take the derivative of the outer function and multiply it by the derivative of the inner function.
For example, with the given function f(x) = \(x+8\)^{1/2}, we consider \(x+8\) as the inner function and \((...)^{1/2}\) as the outer function. First, you would differentiate \((u)^{1/2}\) as if u were just a variable, and then multiply by the derivative of u, which is \(x+8\) in this case. This application leads to an understanding of how slight changes in x affect the function's output.

Differentiation Techniques

Differentiation techniques encompass a variety of methods used to find derivatives of functions. Beyond the chain rule, these include the power rule, product rule, quotient rule, and more advanced techniques such as implicit differentiation and logarithmic differentiation. Each technique is suited to different kinds of functions or situations. For example, the power rule applies to functions with powers of x, while the product rule is used when differentiating the product of two functions.
Students can often simplify a function before applying these rules. Rewriting a function in a more differentiation-friendly form, like turning a square root into a fractional power, is an essential step to make the problem more tractable, as seen in the solution above.

Composite Functions

A composite function is essentially a function within another function, where the output of one function becomes the input of another. In mathematical terms, if g(x) and h(x) are functions, then the composite function f(x) = h(g(x)) is formed by substituting g(x) into h.
The function given in our exercise, f(x) = \(x+8\)^{1/2}, is an example of a composite function. Here, the inner function is g(x) = x+8, and the outer function is h(u) = u^{1/2}, where u=g(x). Understanding the composition of functions is crucial for correctly applying the chain rule.

Power Rule

The power rule is a basic technique in differentiation that is used when you're dealing with powers of x. It states that if you have a function of the form f(x) = x^n, where n is any real number, the derivative of f with respect to x is f'(x) = nx^{n-1}. It's a straightforward rule that's often one of the first ones learned.
In our exercise, we needed to transform the square root into a power to utilize this rule effectively. As a result, the function f(x) = \(x+8\)^{1/2} became easier to differentiate using the power rule along with the chain rule.

Derivative of Square Root Functions

Derivatives of square root functions require special consideration, as they may not be initially set up for direct application of common differentiation rules. By rewriting the square root as a fractional exponent, we prepare it for the power rule. In the problem at hand, f(x) = \(x+8\)^{1/2}, after rewriting it as f(x) = (x+8)^{1/2}, we apply the chain rule in conjunction with the power rule. The outside function is the square root, equivalent to raising to the power of 1/2, and the derivative follows as f'(x) = \(1/2\) \((x+8) \)^{-1/2}, demonstrating the technique for differentiating square root functions.