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

Question

Compute the derivative of $$ y(t)=1+2 t-5 t^{7}+2 e^{3 t}-\ln 6 t+2 \sin t-3 \cos t $$

Step-by-Step Solution

Answer

The derivative is \(y'(t) = 2 - 35t^6 + 6e^{3t} - \frac{1}{t} + 2\cos(t) + 3\sin(t)\).

Step 1: Identify Each Term and Its Derivative Type

To compute the derivative of the given function, break it down into individual terms. The function is: \[y(t) = 1 + 2t - 5t^7 + 2e^{3t} - \ln(6t) + 2\sin(t) - 3\cos(t)\]. Identify the derivative rules needed for each term.

Step 2: Apply the Derivative Rules to Each Term

Differentiate each term using the appropriate differentiation rules: - The derivative of a constant (1) is 0.- The derivative of \(2t\) is 2.- The derivative of \(-5t^7\) is \(-35t^6\) (using the power rule).- The derivative of \(2e^{3t}\) is \(6e^{3t}\) (using the chain rule).- The derivative of \(-\ln(6t)\) is \(-1/t\).- The derivative of \(2\sin(t)\) is \(2\cos(t)\).- The derivative of \(-3\cos(t)\) is \(3\sin(t)\).

Step 3: Combine the Derivatives

Now, combine all the derivatives from Step 2: \[y'(t) = 0 + 2 - 35t^6 + 6e^{3t} - \frac{1}{t} + 2\cos(t) + 3\sin(t)\] Simplify the expression if needed.

Step 4: Simplify the Final Expression

The final expression doesn't need significant simplification, as it's already in its simplest form. Thus, the derivative is: \[y'(t) = 2 - 35t^6 + 6e^{3t} - \frac{1}{t} + 2\cos(t) + 3\sin(t)\].

Key Concepts

Derivative Rules

Before diving into solving derivatives, understanding the different derivative rules is essential. These rules are the foundational tools you need to differentiate any kind of function.

• The Constant Rule helps find derivatives of constant values. If you have a constant, its derivative is 0. Simple, right?
• The Constant Multiple Rule states that if you differentiate a constant times a function, you can take the constant out and just differentiate the function. For example, for a function like 2t, the constant 2 stays as it is, while the derivative of t is 1, so you get 2 as the derivative.
• The Sum Rule is all about adding or subtracting functions. You just differentiate each term separately, then add or subtract those derivatives.

Using these rules in combination can help you tackle even the most complex problems easily. Just break them down into manageable pieces.

Power Rule

The Power Rule is a gem for differentiating terms that involve variables raised to a power. If you have a function in the form of \(x^n\), its derivative will be \(nx^{n-1}\). This rule is straightforward and saves a lot of time.Here is a quick example: If you have \(-5t^7\), its derivative becomes \(-35t^6\). See what happened? Just multiply the exponent (7) by the coefficient (-5) to get -35, and then subtract 1 from the exponent.This method works for any power including fractions or negative numbers. Just remember that the power tells you how the term grows. The derivative shows the rate of that growth.

Chain Rule

The Chain Rule is particularly useful when you differentiate composite functions, those function-within-a-function scenarios. To apply the Chain Rule, let's consider a function like \(2e^{3t}\). Here, \(e^{3t}\) is the composite part. The Chain Rule tells us to differentiate the outer function first and then multiply it by the derivative of the inner function.

• Differentiate the outer part: The derivative of \(e^{3t}\) is \(e^{3t}\), the derivative of exponential functions with a base of 'e' doesn't change.
• Differentiate the inner part: The inner function is \(3t\), whose derivative is 3.
• Lastly, multiply them: resulting in \(6e^{3t}\).

The Chain Rule can seem tricky at first, but with practice, it becomes a powerful tool for efficiently handling more complicated differentiation tasks.

Trigonometric Derivatives

Trigonometric functions are everywhere, and knowing how to differentiate them is very handy! Each of these functions has a specific derivative rule.

• The derivative of \(\sin(t)\) is \(\cos(t)\), meaning \(\sin(t)\) turns into \(\cos(t)\).
• Conversely, the derivative of \(\cos(t)\) is \(-\sin(t)\). The negative sign is essential.

For instance, if you have \(2\sin(t)\), its derivative is \(2\cos(t)\). And for \(-3\cos(t)\), it turns into \(3\sin(t)\). Just carry over the constants here, too.Whenever differentiating trigonometric functions, it’s important to remember these rules and always keep an eye out for those negative signs!