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If $ f( x) = x^2/ (1 + x), $ find $ f" (1). $

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01:46

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 2

The Product and Quotient Rules

Derivatives

Differentiation

Missouri State University

Oregon State University

Idaho State University

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.

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hands clear. So when you right here, so we have a pup. Acts is equal to x square over one plus X. We're gonna difference she using the quotient rule to find the first year of it to get one plus x d over d X for X square minus x square d over DX for one plus x. This becomes equal to all over one plus square. This becomes equal to one plus x times two x minus X square times one over one plus x square just equal to two x plus x square over one plus x square. We're gonna find the second derivative Using the closure role, we get one plus x square D over DX go to X plus X square minus two X plus x square Do you over d x one plus x square all over one plus X square needs. Square that again and this becomes equal to one plus x times two plus two x minus two times two X plus x square all over one plus x cute. We just have to plug in X is equal to one. Then we get, uh, off. The second derivative of one is equal to 14 Yeah,

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