Find the derivative of the following function f(x) = ln(1 -...

Find the derivative of the following function f(x) = ln(1 - x) sin^2 x f'(x) = 1/(1 - x) sin^2 x + 2 ln(1 - x) cos^2 x f'(x) = -1/(1 - x) sin^2 x + 2 ln(1 - x) sin x cos x f'(x) = -1/(1 - x) sin^2 x - ln(1 - x) cos^2 x f'(x) = -1/(1 - x) sin^2 x + ln(1 - x) cos^2 x

Transcript

00:01 All right, so for this problem, we're going to find a derivative of sine of x times cosine of 1 minus x squared.

00:07 Okay.

00:08 So, first thing to notice, we've got sine of x, right, times cosine of 1 minus x squared.

00:17 Okay, so this is going to be a product rule.

00:20 So product rule, f is our first thing, so f is sine of x.

00:27 G is our second thing.

00:29 G is cosine of 1 minus x.

00:33 X squared okay and we need f and f prime and g prime right so f prime all right well the derivative sine of x is cosine of x we need g prime all right well this we've got cosine of of of all right well this we've got cosine of and then some more complicated stuff inside okay so we've got a sort of inside part we need to worry about complicated inside part we've got an outside part okay so for this, we're going to have to use chain rule.

01:03 All right.

01:04 So let's look up here.

01:05 Chain rule means we take the derivative of the outside part and just plug in the inside part.

01:13 And then we multiply by the derivative of the inside part.

01:17 So what this means is, well, our outside part was the cosine.

01:21 So cosine of u is our outside part.

01:25 So if we take the derivative of cosine u, we get negative sine u, we get negative sign.

01:31 U.

01:32 Okay, but we're gonna plug in g of x.

01:34 So we're gonna and excuse me, i say we're gonna plug in g of x, but our gs here are different.

01:40 This is not the same of x.

01:41 We're gonna plug in our inside.

01:43 Okay, we're gonna plug in our inside.

01:44 Okay, and leave our inside the same.

01:45 So negative sign of one minus x squared...

Question

Find the derivative of the following function f(x) = ln(1 - x) sin^2 x f'(x) = 1/(1 - x) sin^2 x + 2 ln(1 - x) cos^2 x f'(x) = -1/(1 - x) sin^2 x + 2 ln(1 - x) sin x cos x f'(x) = -1/(1 - x) sin^2 x - ln(1 - x) cos^2 x f'(x) = -1/(1 - x) sin^2 x + ln(1 - x) cos^2 x

Please give Ace some feedback