Problem 2 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Differentiate (means compute the derivative of) \(P\). Use one rule for each step and identify the rule as, C (Constant Rule), \(t^{n}\left(t^{n}\right.\) Rule), S (Sum Rule), CF (Constant Factor Rule), PC (Power Chain Rule), or E (Exponential Rule). For example, $$ \begin{aligned} \left[\pi t^{-2}-5\left(e^{t}\right)^{7}\right]^{\prime} &=\left[\pi t^{-2}\right]^{\prime}-\left[5\left(e^{t}\right)^{7}\right]^{\prime} \\ &=\pi\left[t^{-2}\right]^{\prime}-5\left[\left(e^{t}\right)^{7}\right]^{\prime} \\\ &=\pi \times(-2) t^{-3}-5\left[\left(e^{t}\right)^{7}\right]^{\prime} \\ &=-2 \pi t^{-3}-5(7)\left(e^{t}\right)^{6}\left[e^{t}\right]^{\prime} \\ &=-2 \pi t^{-3}-35\left(e^{t}\right)^{6} \times e^{t} \\ &=-2 \pi t^{-3}-35\left(e^{t}\right)^{7} \end{aligned} $$ a. \(\quad P(t)=5 t^{2}+32 e^{t}\) b. \(\quad P(t)=3\left(e^{t}\right)^{5}-6 t^{5}\) c. \(\quad P(t)=7-8\left(e^{t}\right)^{-1}\) d. \(\quad P(t)=\frac{2}{5}+\frac{t}{3}\) e. \(\quad P(t)=t^{25}+3 e\) f. \(\quad P(t)=\frac{4}{e}+\frac{t^{5}}{7}\) g. \(\quad P(t)=1+t+\frac{t^{2}}{2}-e^{t}\) h. \(P(t)=\frac{\left(e^{t}\right)^{2}}{2}+\frac{t^{3}}{3}\)

Step-by-Step Solution

Answer

Apply Sum, Power, Exponential Rules, and differentiate term by term to find derivatives.

Step 1: Differentiate Each Term (S - Sum Rule)

According to the Sum Rule, differentiate each term in the expression for the function separately. For all parts, identify the individual functions to differentiate.

Step 2: Apply the Power Rule and Constant Factor Rule

The Power Rule states that \( \left(t^n\right)' = nt^{n-1} \). This can be combined with the Constant Factor Rule (CF), which allows constant factors to remain multiplied. Apply these rules to the \(t^n\) terms in each function.

Step 3: Differentiate Exponential Functions (E - Exponential Rule)

For exponential functions of the type \(e^t\), their derivative is simply themselves: \( \left(e^t\right)' = e^t \). For expressions like \( \left(e^t\right)^n \), use the Power Chain Rule (PC): \( n \left(e^t\right)^{n-1} e^t \). Apply these rules to the exponential terms in each function.

Step 4: Calculate Constants (C - Constant Rule)

The derivative of a constant is zero. Identify all constant terms and note their derivatives for each function in the problem.

Key Concepts

Power Rule

The Power Rule is one of the fundamental rules in differentiation, mainly used when dealing with expressions involving powers of variables. The rule essentially tells us how to differentiate a variable raised to an exponent. In mathematical terms, if you have a term like \( t^n \), its derivative is given by \( n \times t^{n-1} \). This means you take the exponent \( n \), multiply it by the coefficient in front of the term (which is one if there is no coefficient visible), and decrease the exponent by one.

This rule is highly efficient for differentiating polynomial terms. For example, if you're differentiating a term \( 5t^3 \), you'd apply the Power Rule to get \( 3 \times 5 \times t^{3-1} = 15t^2 \). By understanding and applying the Power Rule, you simplify a significant portion of calculus problems involving polynomials.

Remember, the Power Rule only applies to expressions of the form \( t^n \) where \( n \) is a real number, including negative integers.

Sum Rule

The Sum Rule is used when you have to differentiate an expression that is the sum of multiple terms. According to the Sum Rule, the derivative of the sum of functions is the same as the sum of the derivatives of each function. This means you can find the derivative of each term separately and add them together.

For example, if \( P(t) = t^2 + 3e^t \), the Sum Rule allows you to find the derivative of \( t^2 \) and \( 3e^t \) separately, and then add them together. So, you would calculate \( (t^2)' = 2t \) and \( (3e^t)' = 3e^t \), giving you \( 2t + 3e^t \) as the derivative.

This rule is simple yet powerful because it reduces complex-looking expressions into more manageable parts, allowing you to focus on one piece at a time.

Exponential Rule

Differentiating exponential functions, especially those involving the natural exponential base \( e \), is made straightforward by the Exponential Rule. This rule states that the derivative of \( e^t \) is simply \( e^t \).

When dealing with expressions like \( (e^t)^n \), you must apply the Power Chain Rule, which slightly modifies the Exponential Rule. This requires you to treat the exponent \( n \) as a constant multiplier. Therefore, the derivative is \( n(e^t)^{n-1}e^t \).

Imagine a function like \( (e^t)^3 \). You would calculate its derivative as \( 3(e^t)^2e^t = 3(e^t)^3 \). It's crucial to remember that the exponential function \( e^t \) is unique because its derivative is the same as the function itself, making calculations much easier.

Constant Factor Rule

The Constant Factor Rule is a useful tool in calculus, especially when combined with other differentiation rules. If you have a constant multiplied by a function, like \( c \times f(t) \), the derivative is \( c \times f'(t) \). This rule allows the constant to "pass through" the differentiation process, meaning you don't need to change it.

For instance, consider \( 3t^4 \). Applying the Power Rule first gives the derivative of \( t^4 \) as \( 4t^3 \). The Constant Factor Rule lets the coefficient \( 3 \) remain, leading to a derivative of \( 3 \times 4t^3 = 12t^3 \).

This rule is particularly handy for simplifying the differentiation of terms containing constants, avoiding unnecessary complexity in calculations. It's essential to apply the Constant Factor Rule before applying specific derivative rules to functions in your expression. Overall, this rule is crucial for maintaining clarity and simplicity in differentiation.