1. Find f'(x) and f'(1) for each function. Round f'(1) to...

1. Find f'(x) and f'(1) for each function. Round f'(1) to two decimal places. a. f(x)=sin x^5 b. f(x)=sin^5 x c. f(x)=cos (4x+9) d. f(x)=cos^2(x^2+3) e. f(x)=5 e^x f. f(x)=e^{5x} g. f(x)=ln (x+3) h. f(x)=x ln (x^2+4) i. f(x)=3^x j. f(x)=ln sin x k. f(x)=sin ln x l. f(x)=(?x)(3^x)(tan x) m. f(x)=log x ln x n. f(x)=cos (e^{cos x})

Transcript

0:00 All right, hello.

00:01 In this video, i'm going to go through the first seven of these problems to do the last seven just for time's sake.

00:07 If you could post it and say just do, what, h through n, that would be great.

00:11 For these first seven, what we're asked to do here is we're given f of x, and we're going to find f prime of x and f prime of 1 rounded to two decimal places.

00:18 And all of these are functions where we have lots of chain rules, and so we'll go ahead and just explain how to do all these.

00:25 These.

00:25 So for our first function here, sine of x to the fifth, what we're doing is we're taking the derivative of sine, which we know is cosine, and it's going to be the cosine of what's ever inside.

00:34 But because what's inside is its own function, we also have to take the derivative of that inside.

00:38 So we're going to have to multiply by five x to the fourth in order to do the chain rule there.

00:43 That's the first one.

00:44 I'll go ahead and do f prime one for all of these at the same time.

00:48 For sine to the fifth of x, that's what essentially what we're saying is we have sine of x raised to the fifth power.

00:56 And so if we have something raised to a power, the derivative, we take five times the sine of whatever that, or times that function rather, to the power minus one.

01:05 And then because we have a function inside, we have to take the derivative of that.

01:09 So we'll also have times cosine x there.

01:11 For this one here, we know that the derivative of cosine is negative sine of whatever's inside.

01:18 And then because what's inside has a derivative that's not just 1, we also have to take the derivative of the inside, which is going to be 4.

01:26 For this guy here, we could also write this as a cosine of this function squared.

01:33 And so you have to take the derivative of that squared bit.

01:38 So we're going to have 2 times cosine x squared plus 3 to the 1 power...

Question

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