Let f(x) = sec(x) - x. (a) Find f'(x). f'(x) = (b) Check to...

Transcript

00:01 For this problem, we've been given a function, f of x equals e to the x divided by 2x squared plus x plus 1.

00:09 And we want to find the first derivative.

00:13 Taking a step backwards, you can see that this is a quotient.

00:17 So i'm going to need to use the quotient rule to take our derivative.

00:20 So let's review.

00:22 If i have a quotient, some function g of x divided by h of x, and i want to find the derivative.

00:29 The quotient rule says i take the denominator times the derivative of the numerator, minus the numerator, times the derivative of the denominator, all over the denominator squared.

00:48 That's the quotient rule, so i'll need to use this to find the derivative of f of x.

00:54 So let's apply it.

00:57 I start with my denominator, 2x squared plus x plus 1, times the derivative of the derivative.

01:04 Of my numerator.

01:05 Well, this one's easy.

01:06 The derivative of e to the x is just e to the x minus my numerator times the derivative of my denominator.

01:16 Well, the derivative of the denominator is going to be 4x plus 1, all over my denominator squared.

01:28 Okay, so there's the quotient rule.

01:30 Now let's see if we can pretty this up a little bit, just make it a little bit neater.

01:34 Both terms in my numerator have an e to the x in common.

01:38 So i can factor that out.

01:40 And what i'm left with is 2x squared plus x plus 1 minus 4x minus 1 over my denominator.

01:56 Okay.

01:57 Let's simplify what's in those parentheses.

02:00 My constants cancel...

Question

Let $f(x) = \sec(x) - x$. (a) Find $f'(x)$. $f'(x) = $ (b) Check to see that your answer to part (a) is reasonable by graphing $f$ and $f'$ for $|x| < \frac{\pi}{2}$.

Question

Let $f(x) = \sec(x) - x$. (a) Find $f'(x)$. $f'(x) = $ (b) Check to see that your answer to part (a) is reasonable by graphing $f$ and $f'$ for $|x| < \frac{\pi}{2}$.

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