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(a) If $ F(x) = f(x)g(x). $ where $ f $ and $ g $ have derivatives of all orders, show that $ F" = f" g + 2f'g' + fg". $(b) Find similar formulas for $ F''' $ and $ F^{(4)}. $ (c) Guess a formula for $ F^{(n)}. $

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00:54

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 2

The Product and Quotient Rules

Derivatives

Differentiation

Oregon State University

Baylor University

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.

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.

05:46

(a) If $ F(x) = f(x)g(x). …

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(a) If $F ( x ) = f ( x ) …

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Assuming the first and sec…

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Assume that$$f(1)=…

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Find a general formula for…

Hey, it's Claire. So ending. Read here. So for part A, we're going to use the first turret, the product roll, to find the first trip it is. And then we're going to use the product roll on both products to find the second derivative turns the derivative of G plus f of x times, a second or a bit of G of X, just equal to the second derivative of friends G of X plus two terms. The first derivative of terms derivative of G plus that F G comes the second derivative of G for part B, we have the equation for the second derivative. We're gonna difference you again to get the third derivative we got. Dream the second derivative drag this. And when we simplify it, we get three times the first derivative of the second derivative of G plus the terms such a threat of dirt derivative of G. We're gonna different. She again me yet I have to. The fourth is equal to this becomes equal Thio to the fourth three plus six back to the second derivative G to the second derivative plus four tongues, the derivative of G character of it. If and then for part C, we see that where we're going to recall the 2nd 3rd and fourth derivatives. And we see the coefficient. So for the half, or the coefficient of each derivative for the seconds we have 1 to 1. For the third, we have 1331 And for the fourth, we have one for six for one. So these are the paschal triangles. And you think if I know meal expression, this is our formula to the n minus K terms G to the key.

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