Find the derivatives of the following functions. f(x)=x^2 cosx+x tan(x)

Name: Problem 377, Chapter 3: Calculus Volume 1
Uploaded: 2020-04-27T16:17:54-08:00
Duration: 11 min 11 s
Channel: Brian Ketelobeter
Description: Find the derivatives of the following functions. f(x)=x^2 cosx+x tan(x)

Find the derivatives of the following functions. f(x)=x^2...

Key Concepts

Power Rule

The power rule states that the derivative of x raised to a constant power n is n times x raised to the power (n-1). This rule is essential when differentiating polynomial functions and simplifies the process of finding the derivative of expressions like x².

Product Rule

The product rule is used to differentiate the product of two functions. It states that the derivative of u(x)·v(x) is u'(x)·v(x) + u(x)·v'(x). This concept is crucial when dealing with functions that are products of simpler functions, as seen in the given function where terms like x²·cos(x) require this rule.

Sum Rule

The sum rule in differentiation states that the derivative of a sum of functions is equal to the sum of the derivatives of the individual functions. This rule allows one to differentiate each term separately when a function is expressed as an algebraic sum, simplifying the differentiation process.

Trigonometric Derivatives

Trigonometric derivatives involve finding the derivatives of trigonometric functions such as cosine and tangent. For example, the derivative of cos(x) is -sin(x) and the derivative of tan(x) is sec²(x). Understanding these basic trigonometric derivatives is key when differentiating functions that include trigonometric components.

Transcript

00:02 Okay, we are looking at, sorry, this is problem number 377, and we're given a function, and we're supposed to find the derivative.

00:17 So first off, what i like to do is go through and consider what the rules are, well, to break down f of x in such a way to make the derivative, to make the, to make the competition.

00:32 Easier and sort of pinpoint what the rules are that we need to apply.

00:38 So i noticed that first off, if i take if h or sorry, g of x is equal to x squared kos x and and h of x is equal to x tan x then i have that f of x then i have that f of x is equal to g of x plus h of x so f is a sum here is g of x plus h of x and and so from this, the sum rule tells me that f prime at x is equal to g prime of x plus h prime of x.

01:54 Okay.

01:57 And next, i note that, so this is one, let me just move this over here, the time b.

02:07 And since g of x to calculate the derivative of g of x i noticed that it's the product of two functions so we have g of x g of x is equal to r of x times s of x okay and and the product rule then tells me that g prime of x is equal to the first, the derivative of the first times the second, plus the first times the derivative of the second.

03:03 Okay um so we have that guy and finally of course in the h prime very similarly is a product rule it's h is a product itself so we get um h of x uh is equal to r of x times s of x again, then we get that h prime at x is equal to just repeating the same rule again.

03:50 The derivative of the first times the second plus the first times the derivative of the second.

04:01 Okay, so now all we have to do then the next step is to go through and find the corresponding elements here.

04:13 So in particular in g of x case, so we know that let's see.

04:22 So we know that little r of x is equal to x squared.

04:38 So we get r prime of x, of course, is equal to 2x.

04:48 And little s of x is, so little s of x is, so little s of x.

04:55 Here is cosine x so we have little s of x is equal to cosine x and s prime of x of course is equal to negative sign okay so um so then we're computing g prime now let's just plug this guy in okay so yeah, so we're just going to plug this into g prime, our rule for g prime, and we get that g prime at x is just equal to, well, our prime here i say is 2x and times the s of x, which is kos x.

05:53 And then of course and then in the second one it's r of x which of course is x squared times the s prime which is negative sign x okay so this is g prime okay we'll just move uh just move that over to the new screen so i'm just going to i want to note, of course, before i begin on this one, i just want to mention, of course, again, just reiterating the f prime, that's x, is equal to just g prime, that's x, plus h prime of x.

06:54 Okay um so we'll move we'll just move this oops okay so we'll move this right down here um and just flip this around and now we want to do the same thing with eight um so we plug in then uh h prime of course um of course we know.

07:27 Oh, i see.

07:33 I made a mistake.

07:34 Okay.

07:35 Sorry, i jumped ahead of myself...

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