91Ó°ÊÓ

The first derivative of a function tells us about the rate at which the y-value changes as the x-value changes. Essentially, it describes the slope of the tangent line to the curve at any given point. For the curve \(y=(x-2)^{1/3}\), finding the first derivative allows us to understand how the curve rises or falls.

To find the first derivative of \(y=(x-2)^{1/3}\), we use the power rule. The power rule is a basic differentiation rule that states if \(y = x^n\), then \(y' = nx^{n-1}\). Applying this rule to our function, we have:

• The exponent \(n = 1/3\)
• Using the power rule, \(y' = \frac{1}{3}(x-2)^{-2/3}\)

This result tells us that small changes in \(x\) lead to changes in \(y\) that are determined by \(\frac{1}{3}(x-2)^{-2/3}\). It gives us the slope of the tangent at any point on the curve.

The second derivative provides insight into the curvature of the function. This means it tells us whether the function is concave up (curving upwards) or concave down (curving downwards) at any given point. It is also used to find inflection points, where the concavity of the function changes.

For our function \(y=(x-2)^{1/3}\), we found the second derivative using differentiation once more:

• Starting from \(y' = \frac{1}{3}(x-2)^{-2/3}\), we differentiate again to find \(y''\).
• Using the chain rule, the calculation gives us \(y'' = -\frac{2}{9}(x-2)^{-5/3}\).

Examining this second derivative reveals that it is always negative because the expression \( (x-2)^{-5/3} \) is positive, while the coefficient \(-\frac{2}{9}\) is negative. This constancy means the curve does not shift from being concave up to concave down or vice versa; thus, no inflection points exist on the curve.

The chain rule is a fundamental technique in calculus used for differentiating compositions of functions. It is crucial when functions inside functions (nested functions) come up.

In the exercise, the chain rule was used while finding the second derivative of \(y=(x-2)^{1/3}\). Specifically, the derivative of a function like \((x-2)^{-2/3}\) involves identifying an "inner" function \(x-2\) and an "outer" function, here a power function of \(t^{-2/3}\) where \(t = x-2\).

• The derivative of the outer function with respect to the inner is \(-\frac{2}{3}t^{-5/3}\).
• The derivative of the inner function, \(x-2\), with respect to \(x\) is just 1.

Applying the chain rule, we multiply these two derivatives to obtain the second derivative \(y'' = -\frac{2}{9}(x-2)^{-5/3}\). This simplification helps us understand how the rate of change of the slope itself changes.