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Write the composite function in the form $ f(g(x)). $ [Identify the inner function $ u = g(x) $ and the outer function $ y = f(u). $ ] Then find the derivative $ dy/ dx. $$ y = (2x^3 + 5)^4 $

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$f(x)=[g(x)]^{4} \quad g(x)=\left(2 \cdot x^{3}+5\right) \quad \frac{d y}{d x}=24 \cdot x^{2} \cdot\left(2 \cdot x^{3}+5\right)^{3}$

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Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 4

The Chain Rule

Derivatives

Differentiation

Harvey Mudd College

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.

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In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

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Write the composite functi…

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Write the composite func…

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here we have a composite function. So let's start by identifying the inside function. Inside the parentheses, we have two x cubed plus five. So we're going to call that G of X. And now let's identify the outside function. So the inside is being raised to the fourth power. So we're going to say the outside function f of X is X to the fourth power. Okay, Now we want to use the chain rule to find the derivative of this function do y DX. So we start by taking the derivative of the outside function. And so we bring down the four and we raised the inside to the one less power, so that would be to the third. Now we multiply by the derivative of the inside, and the drift of of the inside would be the derivative of two x cubed plus five. And that is six x squared. Okay, Now, if we want a simplify our answer, what I would typically do is multiply my constant four and my other term six x squared, and that would give me 24 x squared times a quantity two x cubed plus five quantity cubed. And there we have our derivative

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