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$f(x)=\left(x^{2}+3 x+1\right)^{17},$ find $f^{\prime}(x)$.

$$17(2 x+3)\left(x^{2}+3 x+1\right)^{16}$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

Campbell University

Oregon State University

University of Michigan - Ann Arbor

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.

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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00:43

f(x)=(x+3)^2-1

All right. So we're gonna find the derivative of this function, and we're gonna use the chain role function and then, uh, we can't do this step by step. You take the derivative of the outer function, you leave the inter function alone and then you multiply by the derivative of the inter function eso. Let's break this down for us, the inter function would be with the parentheses in their function and then the outer function Is this taken to the 17th power eso When we take the derivative, I hope you're ready for this is you do the derivative of the outer function, which is we're used to taking a 17 and moving it to the front. And then it's to the 16th power. Well, what you want to do is leave the inter function alone. This is part of the Chambal, and then you multiply by the derivative of the inside. Well, the derivative of the inside would be the quantity, because you do have to have parentheses to X. The derivative of three X is three and the derivative of 10 So we don't have to write that, um and you can leave your answer like this. This is the change rule. Uh, if you want, multiplication is communicative. So you could move that to express three in front. I believe the answer key does that, but there's really no reason to simplify this Any further were good enough right here.

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