91Ó°ÊÓ

The chain rule is a fundamental tool in calculus for finding the derivative of composite functions. In simpler terms, it helps us differentiate a function that is nested inside another function. For instance, in our exercise, the function is \(y = (x^2 - 5)^{10}\).

To apply the chain rule, we need to identify two parts:

• The outer function: \u = v^{10}\
• The inner function: \v = x^2 - 5\

Using the chain rule, we have to differentiate the outer function and multiply it by the derivative of the inner function. In this case:\[\frac{dy}{dx} = 10(x^2 - 5)^9 \cdot \frac{d}{dx}(x^2 - 5)\]Next, compute the derivative of the inner function:\[\frac{d}{dx}(x^2 - 5) = 2x\]Therefore, the first derivative is:\[\frac{dy}{dx} = 10(x^2 - 5)^9 \cdot 2x = 20x(x^2 - 5)^9\]

The product rule is another essential derivative rule which we use when differentiating products of two functions. It states that the derivative of a product of two functions is given by:

\[ (uv)' = u'v + uv'\]Applying this to our function's first derivative \(20x(x^2 - 5)^9\), let \u = 20x\ and \v = (x^2 - 5)^9\.

We need to find the derivatives of both parts:\[ \frac{d}{dx}(20x) = 20\]\[ \frac{d}{dx}((x^2 - 5)^9) = 9(x^2 - 5)^8 \cdot 2x = 18x(x^2 - 5)^8\]Using the product rule, we combine these results:\[ \frac{d}{dx}(20x \cdot (x^2 - 5)^9) = 20 \cdot 18x(x^2 - 5)^8 + 20x \cdot 9(x^2 - 5)^8 \cdot 2x \]

This simplifies to:\[ 360x(x^2 - 5)^8 + 20(x^2 - 5)^9 \]

The first derivative of a function gives us the rate at which the function is changing. Essentially, it tells us the slope or steepness of the function at any given point. For our function \(y = (x^2 - 5)^{10}\), we found the first derivative using the chain rule.

Let's recap the key steps:

1. Identify the outer and inner functions.
2. Differentiate the outer function: \10(x^2 - 5)^9\.
3. Multiply by the derivative of the inner function: \2x\.

Putting it all together, we get:\[ \frac{dy}{dx} = 10(x^2 - 5)^9 \cdot 2x = 20x(x^2 - 5)^9 \]

This straightforward process makes finding the first derivative easier. It's an essential step before moving to higher-order derivatives, like the second derivative, which provides even more information about the function's behavior.