Problem 10 - Calculus for Biology and Medicine

Question

Differentiate the functions given in Problems with respect to the independent variable.$$ h(t)=\frac{1}{2} t^{2}-3 t+2 $$

Step-by-Step Solution

Answer

The derivative of the function \( h(t) \) is \( h'(t) = t - 3 \).

Step 1: Identify the Function to Differentiate

We are given the function \( h(t) = \frac{1}{2} t^2 - 3t + 2 \). Our task is to find its derivative with respect to \( t \).

Step 2: Apply the Power Rule

The power rule states that the derivative of \( t^n \) is \( nt^{n-1} \). Apply this to each term separately. The derivative of \( \frac{1}{2}t^2 \) is \( 2 \times \frac{1}{2}t^{2-1} = t \).

Step 3: Differentiate the Linear Term

For the linear term \(-3t\), the derivative is simply \(-3\). This follows from applying the power rule where \( n = 1 \).

Step 4: Differentiate the Constant Term

The derivative of a constant like \( 2 \) is \( 0 \). Constants drop out when differentiating as their rate of change is zero.

Step 5: Combine the Derivatives

Now combine all the derivatives: the derivative of the function \( h(t) \) is \( h'(t) = t - 3 + 0 \), which simplifies to \( h'(t) = t - 3 \).

Key Concepts

Power Rule

The power rule is a fundamental tool in calculus for finding derivatives. It states: if you have a term in the form of \( t^n \), then the derivative of this term is \( nt^{n-1} \). This is particularly useful when dealing with polynomials, where each term can be independently differentiated using this rule. For example, when applying the power rule to the term \( \frac{1}{2}t^2 \), you treat \( n \) as \( 2 \). This means:

• Multiply the exponent by the coefficient: \( 2 \times \frac{1}{2} = 1 \).
• Decrease the exponent by 1: \( t^{2-1} = t^1 \).

Therefore, the derivative is simply \( t \). This rule works for any real number \( n \) and can be applied to each term in a polynomial separately.

Derivative of Polynomial

A polynomial is a sum of terms, each consisting of a coefficient and a variable raised to an exponent. Differentiating a polynomial is straightforward, as we can apply the power rule to each term. Consider a typical polynomial expression, such as \( h(t) = \frac{1}{2} t^2 - 3t + 2 \).

• First, apply the power rule to \( \frac{1}{2}t^2 \), giving \( t \) as its derivative.
• Next, consider the term \(-3t\). Using the power rule where the exponent is \( 1 \), it reduces to simply the coefficient: \(-3\).
• The constant \(2\) has a derivative of \(0\), as constants do not change with respect to the variable.

Combining these, the entire derivative for the polynomial is \( t - 3 \), consolidating the derivatives of each individual term.

Differentiation Steps

The process of differentiation involves applying certain rules to find the rate of change of a function. Here's a structured approach to differentiate a polynomial like \( h(t) = \frac{1}{2} t^2 - 3t + 2 \):

1. **Identify the function**: Start with the function you need to differentiate.
2. **Apply the power rule**: Use it on terms with variables. The term \( \frac{1}{2}t^2 \) differentiates to \( t \).
3. **Differentiate linear terms**: For a term like \(-3t\), the derivative is the coefficient \(-3\).
4. **Handle constants**: Any constant term, such as \(2\), has a derivative of \(0\).
5. **Combine the results**: Sum up all the derived terms to get the complete derivative, which for this function is \( t - 3 \).

These steps simplify the process and ensure that no part of the function is overlooked during differentiation. Applying these systematically helps solve more complex functions efficiently.