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Find $$\frac{d^{99}}{d x^{99}}(\sin x)$$

$\frac{\mathrm{d}^{99}}{\mathrm{d} x^{99}} \sin x=-\cos x$

Calculus 1 / AB

Chapter 3

Derivatives

Section 3

Basic Differentiation Formulas

Oregon State University

Harvey Mudd College

University of Nottingham

Boston College

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

03:22

Find $d^{999} / d x^{999}(…

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Find $y^{\prime}$.$$

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01:49

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01:27

we were asked to find the 99th derivative with respect two x of sign of X. So first, let's just recall it. We have the zero with derivative with respect to X of sin of X. This is the same as sign of X itself the first derivative of sign of X. This is co sin of X theseventies derivative with respect to X of sin of X. This is the derivative with respect to X of the first derivative of sign of X. So the derivative of co sin of X, which is negative sign of X and we have that the third derivative of sign of X. Well, this is the derivative of the second derivative of sign of X, which was that's right negative sign of X which is negative times the derivative of sin of X, which is co sin of X and then finally the fourth derivative of sign of X. This is the derivative of negative co sin of X, which is the same thing as the opposite of the derivative of co sin of X, which is negative, Negative sign of X or just sign of X. And so we see that After taking the fourth derivative, we ended up right back where we started with our function Sign of X So you can prove by induction on your own that if N is equal to for K for some integer positive integer here k then we have that the derivative of X sorry derivative of Synnex Respect to X to the end of power I'm this is going to be equal to you. See, in the case K equals one. This is just sign of X again and so it's reasonable to say this is just sign of X. Likewise, if n is of the form four k plus one for some positive into jerk a, then we have that the 10th derivative of sign of X. Well, this is the same as the derivative of sign of X, which is co sin of X. If N is of the form four K plus two for some positive integer, then the end. The derivative of sign of X is the same as the second derivative of sign of X, which we saw above. This is just negative sign of X and finally, if n is of the Form four k plus three for some positive into jerk A. When I say positive into your I am including zero then theme anther derivative of sign of X is the same as thief third derivative of sign of X, which we saw above was negative co sin of X And we see that these cover all possible can all possible ends so and has to be of the form four k four k +14 K plus two or four K plus three. Now notice that 99 is equal to 96 plus three which is four times this is 24 plus three. So we have that are positive integer K in this case is 24 and therefore it follows that the 99th derivative of the sign of X is going to be well. We saw that when we have that 90 nines of the Form four K plus three, this is going to be negative co sin of X. And so this is our answer

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