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Given $H(x)=\left\{5\left(x^{2}-4\right)^{3}+\left(x^{2}-4\right)^{10}\right\}^{1 / 2} ;$ (a) Rewrite $H(x)$ in the form $f g(h(x))) .$ (b) Find the derivative, (c) Find a general rule for the derivative of this, so-called, triple composition.

(a) $f(x)=x^{1 / 2}, g(x)=5 x^{3}-x^{10}, h(x)=x^{2}-4$(b) $5 x\left\{5\left(x^{2}-4\right)^{3}+\left(x^{2}-4\right)^{10}\right)^{-1 / 2}\left(3\left(x^{2}-4\right)^{2}+2\left(x^{2}-4\right)^{9}\right)$(c) $f^{\prime}\left(g(h(x)) g^{\prime}(h(x)) h^{\prime}(x)\right.$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

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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 this problem we'll have H of X Which equals five times x squared minus four cubed Plus expert might score to the 10th to the one half. So we want to rewrite it in the form of F of G of H of X. So the way that we're going to do this is we want F of X to be our outermost function. That's going to be X to the 1/2. And then we're going to have the next outermost function to be our five X cubed -X to the 10th. So we'll have GFX is five x cubed -X to the 10th. And then lastly, we want to choose what's in the Princes which is X squared minus four. That's gonna be our H of X X squared minus four. So this is going to be our final answer.

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