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If $f(x)=a_{5} x^{5}+a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x+a_{0},$ find $f^{(n)}(x),$ where $n$is any positive integer.

$$\begin{array}{l}5 a_{5} x^{4}+4 a_{4} x^{3}+3 a_{3} x^{2}+2 a_{2} x+a_{1} ; 20 a_{5} x^{3}+12 a_{4} x^{2}+6 a_{3} x+2 a_{2} \\60 a_{5} x^{2}+24 a_{4} x+6 a_{3} ; 120 a_{5} x+24 a_{4} ; 120 a_{5} ; \text { the rest are all } 0\end{array}$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

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.

01:15

If $ f(x) = x^5 + x^3 + x …

07:27

If $$f(x)=x^{5}+x^{3}+…

01:47

If $f(x)=2 x^{5}-4 x^{3}+…

03:24

If $f(x)=e^{3 x^{2}+5},$ t…

03:04

(a) Find $f^{(n)}(x)$ if $…

02:19

If $f(x)=x^{5}+x^{3}+x,$ f…

01:19

Let $f(x)=2 x^{3}+5 x+3 .$…

question. Eight states that if f of X equals 85 x to the fifth plus a four x to the fourth, plus a three x to the third, plus a two X squared plus a one X plus a not finding answering motive of F of X, where n is any positive integer. So starting off, we're just gonna take ah, the derivatives until we get to zero. So F prime of X is equal to five a five x to the fourth plus four a four x to the third plus three a three x squared plus two a two X plus eight one second derivative would be 28. 5 x to the third plus 12 a four X squared plus six a three x plus two a two. They're a derivative of X would be 68. 5 X squared plus 24 a four x plus six. A three fourth derivative of X would be 1 28. 5 X plus 24. A four fifth derivative of X would be 1 20. A five six derivative of X is equal to zero so f to the end FX, or the and derivative of F of X is equal to zero for all and greater than or equal to six, and that is your answer to question eight.

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