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Problem 2 Easy Difficulty

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 $

Answer

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

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Video Transcript

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