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Find expressions for the first five derivatives of $ f(x) = x^2e^x. $ Do you see a pattern in these expressions? Guess a formula for $ f^{(n)}(x) $ and prove it using mathematical induction.

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00:57

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 2

The Product and Quotient Rules

Derivatives

Differentiation

Adam R.

March 5, 2019

Oregon State University

Baylor University

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.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

08:48

Find expressions for the f…

02:49

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04:20

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01:20

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03:22

Calculate enough derivativ…

01:07

you know it's clear. So when you raid here, so we have f of X is equal to x square times Ito X. When we differentiate for the first derivative, we apply the product rule and we get eat the X times X square plus two x For a second derivative, we get eat to the axe X square plus four X plus two. And for the third derivative, we eat the X X squared plus six X plus six. For the fourth derivative we could eat the X comes X square plus eight X plus 12. And for the 50 riveted, we got beat to the ex X Square plus 10 X plus 20. And if we see here, we see that the second coefficient is a function of end. So we get our function, which is eat the x X square close to an ex plus and times and minus one. Now, to prove this, we're going to make first unequal one. So this equals just the derivative, the first derivative. So one equation is satisfied. Now we're gonna look for n as equal to K, and we get f is equal to eat the axe times X square plus two k x plus K times came minus one. And finally we're going to proven as equal to K plus one. And this is equal Thio. Hey, the derivative which is equal to eat The X Times X squared plus two k x plus k times came minus one plus e to the X terms to X plus two k Ms is equal to eat to the ex times X square close two times K plus one ex plus que plus one cake So we have proved our formula.

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