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$y=\left(5 x^{2}-7 x+2\right)^{3 / 4},$ find $d y / d x$.

$$\frac{3(10 x-7)}{4\left(5 x^{2}-7 x+2\right)^{1 / 4}}$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

Campbell University

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University of Michigan - Ann Arbor

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.

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.

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03:30

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01:38

Given $2 y^{2}-5 x^{4}-2-7…

so we're tasked with finding the derivative of this function. It's five x squared, minus seven X plus two and it's all to the three force power. So what I like to do is look for parentheses and identify that we have this inter function and this outer function eso when we do the chain room, which is exactly what we're ready for. Uh, we can recognize that. Okay, well, we're going to do the derivative of the outer function first. So what that looks like is removed the exponents in front the three force and then we subtract one from that exploded. Now, remember, subtracting one would be like subtracting four force, so that's equal to negative. 1/4 is your new exponents, and what you'll see is that you leave the inside alone. I've X squared nine and seven X plus two, but then the general says you have to multiply by the derivative of the inside function. Now it's the quantity of the directive of five X, where does 10 X and the derivative of Mega seven. X is native seven and then the derivative of 20 so we don't have to write that. So now uh, some teachers actually let you leave your answer like this, but some teachers want to recognize that you understood that three force, Um and then the 10 X minus seven stays in the numerator, and then that negative power goes back into the denominator. Onda. Some teachers might even ask you to write as the fourth root of five X squared minus seven x plus two, or leave it as to the 1/4 power in the denominator. But however you want to write, or however your professor expects you to write, it is more important. But this is a correct answer. So does this appear, though?

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