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(a) Find $d y$ if $y=\frac{3 x-1}{2 x-3}$,(b) Find $f^{\prime}(x)$ if $f(x)=\sqrt{2 x^{3}+3 x+2}.$

(a) $\frac{-7}{(2 x-3)^{2}} d x$(b) $\frac{3\left(2 x^{2}+1\right)}{2 \sqrt{2 x^{3}+3 x^{2}+2}} d x$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 6

Linearization and Differentials

Derivatives

Campbell University

Oregon State 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.

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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So for the first problem We are given why is he going to three X -1 Over two X -3. And we know it's in the form with you over fee. We are asked to differentiate and find dy so we know that when it's in this form we can use the formula D U V D x. T X is equal to V D U D X minus you. T V TX. All over fi squared. Obviously we need to find the U. T. X. Which is found by differentiating the numerator is equal to three tv T X. By differentiating the denominator which is equal to two. Now plugging in our values into the formula V. Is two X. Okay so now we're looking for dy dx So now V is going to Fee is going to X -3. The U. DX which is three You is the co two, X -1. Apply by DVD X. Which is to All over the square which is two X -3 squared. Simplifying. This will give us six x minus nine minus six x plus two. All Over two X -3. It's squared. Now the 60s council out and we are left with minus seven Over two X -3 squared. But this is dy dx since we want dy G y is equal to multiply both sides by dx to get minus 7/2 x minus three squared dx. His our first solution next we are going to differentiate F X. Which is given as two X cubed Plus three x plus two. Mhm And this can be rewritten as to execute plus three X plus two. To the power of half. No to find F prime of X. We know that by the Parlow. First the power comes down becomes half more. Applied by what's in the brackets returns as it is first. Yeah. Plus to multiply by the differential of what's in the pocket which is which will give a six X squared plus three. Now the power here the power here. Okay, let's do this again. We missed the power. So we're gonna have the park coming down two X squared cubed plus three X plus two. And the power we subtract one which give us minus half. Well played by what's inside the progress give us six X squared plus three, simplifying this. Who give us six X squared plus three All over to who took off Because the -1 makes this um we have to invent this and we get to X cubed plus three X plus two of the denominator. So this is our solution. We can further simplify it by factoring out at three. Here we are left with two X squared plus one. All over two. The hotel two X cubed plus three X. It has to his um f prime of X. And this Is a 2nd solution. Mhm. Mhm mm hmm

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