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Compute $\quad(a) \frac{d}{d y}\left(y^{3}+3 y^{2}+4\right) ; \quad(b) \frac{d}{d x}\left(y^{3}+3 y^{2}+4\right);$ $(c) \frac{d}{d x}\left(y^{3}+3 y^{2}+4\right)^{10}$

(a) $3 y^{2}+6 y$(b) $\left(3 y^{2}+6 y\right) \frac{d y}{d x}$(c) $30 y(y+2)\left(y^{3}+3 y^{2}+4\right)^{9} \frac{d y}{d x}$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

Campbell University

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.

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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Alright, This problem is just based off of this uh why Q plus three y squared plus four eso the first part while we're doing the derivative in terms of why out male, I'll just rewrite it. It's easier that way is you just do this problem as if it's a normal like why is the independent variable? So you bring the three in fronts of track one from the exponents, three attempts to a six. Why, and the director of the 40 So there's part the difference with part B, though. Have you noticed it says dy dx Um So what? That means, uh is that you still take the derivative of that the same way we did. But then what you have to do because we want things in terms of X, then we have to multiply that thing that we just found wait to tell the reader like, Hey, now, why is the function of X um that we need to take the derivative of that piece, whatever that is. Eso Then what's new about the third piece? Part C is what if we're taking the derivative of this function? Believe it's to the 10th Power to the 10th power. Um so a was not the change rule be was the change rule? See, is the chain role inside of the chain rule where you bring the 10 in front, You leave the Y Q plus three y squared, plus four alone. It's now to the ninth Power because you have to subtract one from the exponent. Then you multiply by the derivative of being side. Which is why Q plus sorry not why Cubed did it right to twice before three y squared plus six y, and then you have to multiply by. You have to do what we did in part B when we have to say, Hey, we did the directive. Why? So now it take the derivative. Why, with respect to X now you can simplify that. If you want most, Matthew will let you leave it like that. So there's a B C

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