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

Step 1: Differentiate each term

To find the first derivative of the function, apply the power rule to each term of the function separately. For a function of the form \( g(t) = t^n \), the derivative is \( g'(t) = n \cdot t^{n-1} \). Apply this to the function:- The derivative of \( t^{-5/2} \) is \( -\frac{5}{2} \cdot t^{-\frac{5}{2} - 1} = -\frac{5}{2} \cdot t^{-\frac{7}{2}} \).- The derivative of \( -t^{1/2} \) is \( -\frac{1}{2} \cdot t^{\frac{1}{2} - 1} = -\frac{1}{2} \cdot t^{-\frac{1}{2}} \).

Step 2: Simplify the first derivative

Now, write the first derivative in a simplified form by combining terms from Step 1:\[ g'(t) = -\frac{5}{2} \cdot t^{-\frac{7}{2}} - \frac{1}{2} \cdot t^{-\frac{1}{2}} \].

Step 3: Differentiate each term again

To find the second derivative, apply the power rule again to each term in the first derivative:- The derivative of \( -\frac{5}{2} \cdot t^{-7/2} \) is \( -\frac{5}{2} \cdot (-\frac{7}{2}) \cdot t^{-\frac{7}{2} - 1} = \frac{35}{4} \cdot t^{-\frac{9}{2}} \).- The derivative of \( -\frac{1}{2} \cdot t^{-1/2} \) is \( -\frac{1}{2} \cdot (-\frac{1}{2}) \cdot t^{-\frac{1}{2} - 1} = \frac{1}{4} \cdot t^{-\frac{3}{2}} \).

Step 4: Simplify the second derivative

Write the second derivative in a simplified form by combining terms from Step 3:\[ g''(t) = \frac{35}{4} \cdot t^{-\frac{9}{2}} + \frac{1}{4} \cdot t^{-\frac{3}{2}} \].

Key Concepts

Power Rule

The power rule is a fundamental tool in calculus for finding derivatives. It is especially useful when dealing with polynomials and functions of power form. When you have a function like \( f(t) = t^n \), the power rule allows you to find the derivative quickly: you multiply the exponent \( n \) by the coefficient of \( t \), then reduce the exponent by one. So, applying the power rule to \( f(t) = t^n \) gives the derivative \( f'(t) = n \cdot t^{n-1} \). This method greatly simplifies the process of differentiation.

• For example, to differentiate \( t^{-5/2} \), you multiply \(-5/2\) by \( 1 \) (as it is the coefficient of \( t^{-5/2} \)), resulting in \(-5/2\), and reduce the exponent by 1 to get \( t^{-7/2} \).
• Similarly, differentiating \( -t^{1/2} \) results in \(-1/2 \cdot t^{-1/2}\).

First Derivative

The first derivative of a function represents its rate of change or the slope of the tangent line at any point on the curve. By applying the power rule as discussed above, we can derive the first derivative easily.

For the function \( g(t) = t^{-5/2} - t^{1/2} \), the first derivative involves using the power rule separately for each term:

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

Thus, the first derivative of the function simplifies to:\[ g'(t) = -\frac{5}{2} \cdot t^{-7/2} - \frac{1}{2} \cdot t^{-1/2} \]This derivative tells us how the function \( g(t) \) changes as \( t \) changes.

Second Derivative

The second derivative is essentially the derivative of the first derivative. It gives insight into the concavity of the function and can tell us if the function is curving upwards or downwards at any point.

To find the second derivative of \( g(t) \), you apply the power rule to each term in the first derivative \( g'(t) = -\frac{5}{2} \cdot t^{-7/2} - \frac{1}{2} \cdot t^{-1/2} \):

• The derivative of \( -\frac{5}{2} \cdot t^{-7/2} \) results in \( \frac{35}{4} \cdot t^{-9/2} \).
• The derivative of \( -\frac{1}{2} \cdot t^{-1/2} \) is \( \frac{1}{4} \cdot t^{-3/2} \).

Combine these results to form the second derivative:\[ g''(t) = \frac{35}{4} \cdot t^{-9/2} + \frac{1}{4} \cdot t^{-3/2} \]The second derivative gives important information on the acceleration of the function's rate of change and can be used to identify points of inflection where the curvature type changes.