If we need to calculate the derivative of a function at a point, there are two ways we can think about doingthis. For example suppose f() = x – x, and we need to determine the value of f(4).One option is to just calculate the derivative at that point by plugging the point into the limit definition, likethis:f(4 + h) – f(4)((4 + h)² – (4 + h)) - (4² – 4)f'(4) = lim(16 + 8h + h2 -4 – h) - (16 - 4)lim7h + h2= limh(7+ h)limh= limh→0hlim 7+h = 7.||hhhAnother option is to calculate f'(x) in terms of x, and then plug in a value for x at the end, like this:f(x + h) – f(x)(2+ h) – (x + h)) - (2² – a)(22 + 2xh + h² – z – h) – (22 – 2)limf'(x) = lim= lim2xh + h2 - hh(2x + h - 1)= limh 0hh= lim= limhhhthat is to say, f'(x)= 2x – 1, and therefore f'(4) = 2(4) – 1 = 7.Notice that we get the answer f'(4) = 7 both ways. The second approach might look somewhat morecomplicated at first, but it turns out to be much more efficient if we ever need to know the value of thederivative at multiple points. For example, if we now wanted to know f' (1), f'(7) and f'(11), we wouldn'tneed to calculate the limit definitions all over againformula f'(x) = 2x – 1.-- we could just plug the numbers 1, 7 and 11 into theLet g(x) = x – x².g(3 + h) – g(3)Find g'(3) by calculating limh-0hg(x + h) – g(x)Next, find g'(æ) by calculating limh 0hFinally, find g'(3) by plugging 3 into g'(x).

Name: If we need to calculate the derivative of a function at a point, ther
Uploaded: 2024-03-20T06:47:22.000Z
Channel: Bartleby
Description: If we need to calculate the derivative of a function at a point, there are two ways we can think about doingthis. For example suppose f() = x – x, and we need to determine the value of f(4).One option is to just calculate the derivative at that poin