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Find $\frac{d}{d x}\left(\frac{3}{x^{2}}\right)$ by (a) using the power rule and; (b) using the quotient rule.

$-6 / x^{3}$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 5

Derivative Rules 2

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.

03:09

Find $\frac{d}{d x}\left(\…

02:52

01:44

Find $d y / d x$ by (a) us…

03:35

Use the Quotient Rule to e…

02:07

01:50

(A) Find $f^{\prime}(x)$ u…

01:40

02:12

Let $y=\left(x^{2}+3\right…

01:59

01:57

01:52

02:09

Find each derivative and s…

06:53

Find a formula for the der…

All right, So we're asked to find the derivative of three over X squared on the first way that they want you to do this is to rewrite this problem as three x to the negative second power. And if you do the problem this way, then you bring that expo in front. So we're looking at negative six X to the negative third power as you subtract one from the exponents. So we're looking at negative six over. Execute. Perfect. So that's part a on part B, though What they want you to do is think of this as the quotient rule, which the derivative of the top zero, maybe the bomb alone. Well, zero times anything is nothing. So, you know, that's that's a big fat nothing minus the derivative of the bottom, which is two X. You leave the top alone all over the denominator squared. Well, as we simplify that, you know, switch over toe green again. Zero. This is a zero. So we're looking at negative six x over X to the fourth. Uh, I'll go ahead and show this. We have four x s on bottom so we could cancel out of one of them. What it boils down to is that we have negative six over X cubed. Um, so my Redpath here to grain is part B, whereas my black path appear was part a And no matter what, you get this answer.

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