Y=(2 √(x))/(3(1+√(x)))

Name: Problem 20, Chapter 3: Calculus
Uploaded: 2020-08-02T15:23:23-08:00
Duration: 7 min 2 s
Channel: Sinisa Stura
Description: y=(2 √(x))/(3(1+√(x)))

Key Concepts

Radical Expressions

This concept deals with expressions that include roots, such as square roots, which require the radicand (the expression inside the radical) to meet specific conditions (like being non-negative in the case of real numbers). Understanding radical expressions is essential for manipulating and simplifying functions that include them.

Rational Functions

Rational functions are expressions formed by the ratio of two algebraic expressions. They require careful analysis because the denominator cannot be zero, and their behavior can often be better understood by studying the factors in both the numerator and the denominator.

Domain Restrictions

Domain restrictions involve determining the set of input values for which a function is defined. When a function includes radicals and rational expressions, the conditions for the radicand and the denominator must both be considered to avoid undefined expressions such as negative values under even roots or division by zero.

Simplification of Algebraic Expressions

Simplification involves rewriting algebraic expressions in a more compact or understandable form without changing their value. This can include factoring, canceling common factors, and substituting expressions (like replacing a radical with a variable) to make the analysis or computation of the function more straightforward.

Transcript

00:02 Hello, in this problem we are given y equals a function of x which is two squared of x over three multiplied by one plus square root of x and the question is what is the differential of y or d y equals what we know is that the y equals if derivative of f times d x so basically what the problem wants to do is to solve to find the derivative of this function here okay let's recognize this as a quotient it's u over the u x two functions u and v and the derivative of quotient is by the equation rule, v squared in the denominator and then in the numerator we have the derivative of the numerator times v minus u times v.

01:16 Let's see u is two times the square of x.

01:24 The square of x can be written as x of the power of one half.

01:28 So when we want to differentiate or to find the derivative, we will use the power rule.

01:41 The power rule is, power rule is that x to the power of n.

01:49 The derivative is n times x to the power n minus 1.

01:55 So this is going to be this is a constant multiple so we apply the cost multiple rule is going to be two times the derivative of this and the derivative of that is n n which is one half times x to the power of reduce one half by one minus one half okay so two and two will cancel out and this will be one over x the power of one half or one over x 1 over the square root of alright what is v v is 3 times 1 plus square of x or 3 plus 3 square root of x so v derivative of v is going to be 0 for this part and for this part it's going to be 3 times 1 half same as up here x2 minus 1 half so this is going to be 3 2 and x to the 1 half goes here which is this and now we are ready to find what the derivative of if it is this this is going to be squared in the so this square is 9 and this squared is squared...

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