Repeat Exercise 16, for: (a) f(x)=x^1 / 2 /(x+1), (b) f(x)=(x+1) / x^1 / 2

Name: Problem 17, Chapter 2: Applied Calculus For Business, Economics, and Finance
Uploaded: 2021-02-10T17:46:02-08:00
Duration: 3 min 51 s
Channel: Gregory Higby
Description: Repeat Exercise 16, for: (a) f(x)=x^1 / 2 /(x+1), (b) f(x)=(x+1) / x^1 / 2

Repeat Exercise 16, for: (a) f(x)=x^1 / 2 /(x+1), (b)...

Key Concepts

Power Rule for Derivatives

The power rule is one of the most fundamental differentiation techniques in calculus. It provides a quick method for calculating the derivative of a term of the form x^n, where n is any real number, by multiplying by the exponent and reducing the exponent by one. This rule applies equally to integers, fractions, and negative numbers, making it very useful for handling functions involving radical expressions.

Simplifying and Converting Radicals

Many functions in calculus involve radicals, which are often more easily manipulated when expressed as fractional exponents. Converting between radical notation and exponential notation not only simplifies the differentiation process but also aids in combining terms, applying the power rule, and performing algebraic manipulations required to simplify the final result.

Quotient Rule

The quotient rule is a method in calculus used to differentiate functions that are expressed as the quotient of two differentiable functions. It states that the derivative of a ratio f(x)/g(x) is given by (f'(x)g(x) - f(x)g'(x)) divided by [g(x)]². This rule is essential when dealing with functions presented as fractions, ensuring that the differentiation process accounts for both the numerator and denominator.

Domain Considerations

Functions that include radicals and expressions in the denominator require careful attention to the domain. For radical functions, the expression inside the radical (the radicand) must typically be non-negative, and for rational functions, the denominator must not be zero. Recognizing these domain restrictions is essential for correctly analyzing and interpreting the behavior of the function.

Transcript

00:01 All right, so we're looking at these reciprocal functions, x to the one half over x plus one.

00:09 We're going to do the math there, and then on this side we're going to do x plus one over x to the one half.

00:15 So they're definitely reciprocal functions.

00:18 We're talking about horizontal tangents.

00:21 So when you see that phrase horizontal tangents, you should automatically think, okay, if this is f of x equals, then what we need to do is find the derivative.

00:31 And set that equal to zero.

00:33 And that only happens when the numerator equals zero.

00:38 We don't care about the denominator.

00:42 Yeah, so we're going to do the quotient rule, which the derivative of the top is one -half x to the negative one -half power.

00:53 You leave the bottom alone, minus.

00:57 Now you take the derivative of the bottom, leave the top alone, all over the denominator squared.

01:04 But again, we don't really care about the denominator.

01:08 Because that i mean that can never even equal zero because we're only going to get positive numbers there i guess it could equal zero if x is negative one but anyway we don't care about the denominator so as we examine this problem and we you know distribute in the numerator and we start adding things we have one -half x to the one -half power plus one half x to the negative one half power minus x to the one half power is equal to zero.

01:45 Let's see, one half minus one would be negative one half.

01:56 And let's add that over...

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