1. Higher derivatives: (a) If f(x) = sin(x), find f(16)(x)...

1. Higher derivatives: (a) If f(x) = sin(x), find f(16)(x) and f(59)(x). Do you see a pattern? (b) Find a formula for f(n)(x) given that f(x) = ex sin(x). Do you see how part (a) might be insightful?

Transcript

00:01 In this question we have given f of x is equals to sine x.

00:08 We have to find 16th derivative of f of x and 59th derivative of f of x.

00:18 So for that, first we will consider our function that is f of x is equal to sine x.

00:26 If we differentiate it for the first time then we'll get it as cos x.

00:31 Now if we double differentiate it then we'll get it as minus sine x.

00:38 If we take the third derivative then we'll get it as cosx.

00:46 Now if we take the fourth derivative then we'll get it as sine x.

00:51 So this is the first derivative is same as the fourth derivative.

00:56 Now if we differentiate our fourth derivative then we'll get it as cosx and this is equals to our first differentiation.

01:05 Similarly this is equals to our sixth derivative.

01:09 This will be equal to our seventh derivative.

01:11 This will be equal to our eighth derivative.

01:14 Now we proceed like this then this will equal to ninth derivative, tenth derivative, 11th derivative, 12, 12, 13, 14, 15 and 16.

01:25 So the 16th times derivative of f of x will be sine x only.

01:30 So the answer to our 16th derivative of f of x is equal to sine x now 4 .59th derivative of f of x is equal to now the nearest number to 59 that is divisible by 4 is 56 so 56 time derivative of f of x will be the original function sine x and after this 57th derivative will be cos x 58 will be minus sine x and 59th derivative will be posit...

Question

1. Higher derivatives: (a) If f(x) = sin(x), find f(16)(x) and f(59)(x). Do you see a pattern? (b) Find a formula for f(n)(x) given that f(x) = ex sin(x). Do you see how part (a) might be insightful?

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