Problem 81 - Calculus for Biology and Medicine

Question

Find the first and the second derivatives of each function. \(g(t)=t^{-5 / 2}-t^{1 / 2}\)

Step-by-Step Solution

Answer

The first derivative is \(g'(t) = -\frac{5}{2} t^{-7/2} - \frac{1}{2} t^{-1/2}\), and the second derivative is \(g''(t) = \frac{35}{4} t^{-9/2} + \frac{1}{4} t^{-3/2}\).

Step 1: Find the First Derivative

To find the first derivative of the function, we apply the power rule for derivatives, which states that if you have a function in the form \( f(x) = x^n \), its derivative \( f'(x) \) is given by \( nx^{n-1} \). For \( g(t) = t^{-5/2} - t^{1/2} \),- The derivative of \( t^{-5/2} \) is \( \frac{d}{dt} t^{-5/2} = -\frac{5}{2} t^{-5/2 - 1} = -\frac{5}{2} t^{-7/2} \).- The derivative of \( -t^{1/2} \) is \( \frac{d}{dt} (-t^{1/2}) = -\frac{1}{2} t^{1/2 - 1} = -\frac{1}{2} t^{-1/2} \).Thus, the first derivative is:\[ g'(t) = -\frac{5}{2} t^{-7/2} - \frac{1}{2} t^{-1/2}. \]

Step 2: Find the Second Derivative

To find the second derivative, we will again use the power rule for each term in the first derivative \( g'(t) = -\frac{5}{2} t^{-7/2} - \frac{1}{2} t^{-1/2} \).- The derivative of \( -\frac{5}{2} t^{-7/2} \) is \( -\frac{5}{2} \cdot \frac{d}{dt} t^{-7/2} = \left(-\frac{5}{2}\right)\left(-\frac{7}{2} t^{-7/2 - 1}\right) = \frac{35}{4} t^{-9/2} \).- The derivative of \( -\frac{1}{2} t^{-1/2} \) is \( -\frac{1}{2} \cdot \frac{d}{dt} t^{-1/2} = \left(-\frac{1}{2}\right)\left(-\frac{1}{2} t^{-1/2 - 1}\right) = \frac{1}{4} t^{-3/2} \).Thus, the second derivative is:\[ g''(t) = \frac{35}{4} t^{-9/2} + \frac{1}{4} t^{-3/2}. \]

Key Concepts

Power Rule

The power rule is an essential technique in calculus, particularly in differentiation, which allows us to find derivatives of functions involving powers of variables. The basic idea is that if a function is represented as \( f(x) = x^n \), the derivative, denoted as \( f'(x) \), is found by multiplying the power \( n \) by the variable raised to the power of \( n - 1 \). In formula form, the power rule is expressed as follows:

• \( f'(x) = nx^{n-1} \)

This rule applies to any real number \( n \), including fractions and negative numbers, making it extremely versatile. For instance, if we take \( t^{-5/2} \) as an example from the original problem, using the power rule would mean multiplying \(-5/2\) by \( t\) and reducing the exponent by one. This results in \(-\frac{5}{2} t^{-7/2} \).
This rule simplifies the process of differentiation, allowing us to quickly find the slopes of tangent lines to functions that feature polynomial terms. Practicing with various functions helps students become comfortable and efficient with using the power rule.

First Derivative

The first derivative of a function provides significant insights into the behavior of the function. It essentially measures the rate at which the function's value changes with respect to changes in the input variable. In our example, we are calculating the first derivative of the function \( g(t) = t^{-5/2} - t^{1/2} \).
To do this, we apply the power rule to each term separately:

• The derivative of \( t^{-5/2} \) becomes \(-\frac{5}{2} t^{-7/2} \).
• The derivative of \( -t^{1/2} \) becomes \(-\frac{1}{2} t^{-1/2} \).

Adding these results gives us the first derivative:

• \( g'(t) = -\frac{5}{2} t^{-7/2} - \frac{1}{2} t^{-1/2} \)

The first derivative tells us about the slope of the function at any point \( t \). It indicates whether the function is increasing or decreasing at that point. A positive first derivative suggests the function is increasing, while a negative first derivative suggests it is decreasing. Hence, the first derivative is a fundamental tool for understanding the dynamics of functions.

Second Derivative

The second derivative of a function provides deeper insights into the function's behavior by describing its concavity—the direction in which the function curves. In the context of our problem, after finding the first derivative, we again apply the power rule to find the second derivative.
The steps include:

• Finding the derivative of \(-\frac{5}{2} t^{-7/2} \), which results in \( \frac{35}{4} t^{-9/2} \).
• Finding the derivative of \(-\frac{1}{2} t^{-1/2} \), which results in \( \frac{1}{4} t^{-3/2} \).

Combining these results, the second derivative is:

• \( g''(t) = \frac{35}{4} t^{-9/2} + \frac{1}{4} t^{-3/2} \)

The sign of the second derivative is crucial:

• A positive second derivative suggests the function is concave up, similar to a smiley face, indicating that the rate of increase itself is increasing.
• A negative second derivative suggests the function is concave down, like a frown, indicating that the rate of increase is decreasing.

This understanding helps in analyzing points of inflection where the function changes from concave up to concave down, or vice versa.