Find f^'(x) for the given function f. f(x)=(2 x^2+1)^1 / 3(3 x^3+1)^4 / 3

Name: Problem 74, Chapter 3: Calculus Single Variable
Uploaded: 2022-02-26T18:02:03-08:00
Duration: 7 min 19 s
Channel: Steven Clarke
Description: Find f^'(x) for the given function f. f(x)=(2 x^2+1)^1 / 3(3 x^3+1)^4 / 3

Find f^'(x) for the given function f. f(x)=(2 x^2+1)^1 / 3(3...

Key Concepts

Product Rule

The product rule is used when differentiating a function that is the product of two or more functions. It states that the derivative of a product f(x) = g(x)h(x) is given by f'(x) = g'(x)h(x) + g(x)h'(x).

Chain Rule

The chain rule is applied when differentiating composite functions, where one function is nested inside another. It states that the derivative of a composite function f(g(x)) is f'(g(x)) multiplied by g'(x).

Power Rule

The power rule simplifies the process of differentiating functions of the form x^n, where n is a constant. It states that the derivative of x^n is n*x^(n-1), and this rule is also applied when differentiating functions where the inner function is raised to a power.

Transcript

00:01 Okay, here i want to find f dash of x, differential, and we have here a product.

00:06 So this item will call u, this one will call v.

00:10 And to do it, i'll use the product rule.

00:13 So to work out f dashed of x, it would be u, so 2x squared plus 1, power of 1 third, no change there, times dv by the x, but differentiate v.

00:30 And here i use the chain rule.

00:33 So four thirds goes in front, three x cubed plus one.

00:40 Four thirds minus one is one third.

00:44 And then times that at the differential of what's inside the bracket here, which is nine x squared.

00:54 That's the first part of it.

00:55 That's half of the process.

00:57 Plus, then i would do v.

01:02 So v is 3x cubed plus 1 power of 4 thirds times and then the u by d x differentiate the u function so i get 1 3rd in front and then repeat 2x square plus 1 and 1 3rd minus 1 is negative 2 thirds and then on the chain rule times up by 4x step by step that's the first part of the process.

01:41 For the words, here is u, or this here is v -dashed.

01:48 Here is v, and all this here is u -dashed.

01:54 Now, that's the answer, but i think it's simpler.

01:58 It's just a matter of algebra now.

02:02 So let's take out some factors.

02:05 What we're going to have here, let's take out one -third as a factor to you in.

02:14 I see here we have one -third times four, four -thirds times nine, so one -third i'll take out.

02:26 Then two x squared plus one is a factor.

02:31 And what power do you take out? well the rule is if one's plus and one's minus take out the minus power.

02:39 Here is like one third, here is minus two thirds.

02:43 We'll take out the negative power.

02:45 We'll see why in a moment.