91Ó°ÊÓ

To find the first derivative of a function is to determine how that function changes when the input changes slightly. It examines the slope or the steepness of a curve at any point.
In our context, we have the function \( g(t) = \sqrt[3]{t} \), which we can rewrite as \( g(t) = t^{\frac{1}{3}} \).
Using the power rule, which we will explore in further detail, we find that the first derivative is \( g'(t) = \frac{1}{3}t^{-\frac{2}{3}} \). This result tells us that the rate of change of \( g(t) \) depends on \( t^{-\frac{2}{3}} \), meaning the slope is determined by this expression.
An important application of finding the first derivative is to identify slopes and determine where a function is increasing or decreasing. It provides a crucial stepping stone for further analysis, such as finding the second derivative, to study more complex features of the function like concavity.

The second derivative is a further refinement in understanding the behavior of a function, measuring the rate at which the first derivative itself changes.
For function \( g(t) = \sqrt[3]{t} \), we computed the first derivative as \( g'(t) = \frac{1}{3}t^{-\frac{2}{3}} \). Taking the derivative of this result gives us the second derivative \( g''(t) = -\frac{2}{9}t^{-\frac{5}{3}} \).
This second derivative provides insights on the concavity of the function. If it changes sign, it indicates an inflection point where the graph might bend in the opposite direction.
However, in this particular case, we observe that \( g''(t) \) remains negative, meaning the function does not change concavity.

• This implies that there are no inflection points for the cube root function.
• In practical terms, since the second derivative is always negative, the curve remains concave downwards throughout.

The power rule is a powerful tool in calculus that simplifies finding derivatives of polynomials.
It is particularly useful because it provides a straightforward way to handle powers of variables without complex computations.
Given a function in the form \( f(x) = x^n \), the power rule states that its derivative is \( f'(x) = nx^{n-1} \). This rule applies directly to our example.
We started with \( g(t) = t^{\frac{1}{3}} \), which allowed us to use the power rule to easily arrive at \( g'(t) = \frac{1}{3}t^{-\frac{2}{3}} \).

• This formula tells us precisely how to find the slope of any curve represented by a power function.
• It's beneficial for quickly deriving most polynomial functions, thus enabling swift further analysis with second derivatives and other calculus operations.

Understanding the power rule allows us to tackle a wide array of derivative problems efficiently and accurately.