Problem 35 - University Calculus: Early Transcendentals

Question

Find the derivatives of the function. $$s=2 t^{3 / 2}+3 e^{2}$$

Step-by-Step Solution

Answer

The derivative is \( 3t^{1/2} \).

Step 1: Identify the components of the function

The given function is \[ s = 2t^{3/2} + 3e^2 \]where \( 2t^{3/2} \) is a term with a variable \( t \), and \( 3e^2 \) is a constant term because \( e^2 \) is a constant since \( e \) is the base of the natural logarithm and raised to a constant power.

Step 2: Differentiate each term separately

To find the derivative \( \frac{ds}{dt} \), differentiate each term separately with respect to \( t \). The derivative of a constant is 0, so the derivative of \( 3e^2 \) is 0.Next, apply the power rule \(( \frac{d}{dt}[t^n] = nt^{n-1} )\) to the term \( 2t^{3/2} \).

Step 3: Differentiate \(2t^{3/2}\) using the power rule

Apply the power rule to \( 2t^{3/2} \). First take the derivative of the exponentiated part: \[ \frac{d}{dt}[t^{3/2}] = \frac{3}{2}t^{3/2-1} = \frac{3}{2}t^{1/2} \]Multiply it by the constant, 2:\[ \frac{d}{dt}[2t^{3/2}] = 2 \times \frac{3}{2}t^{1/2} = 3t^{1/2} \]

Step 4: Combine the derivatives

Now, combine the derivatives obtained:The derivative of \( 2t^{3/2} \) is \( 3t^{1/2} \), and the derivative of \( 3e^2 \) is 0.Add them together to get the overall derivative:\[ \frac{ds}{dt} = 3t^{1/2} + 0 = 3t^{1/2} \]

Key Concepts

Understanding the Power Rule in Differentiation

The power rule is a fundamental tool in calculus, especially useful for easily finding derivatives of polynomials and terms involving powers of variables. When you have a term in the form of \( t^n \), the power rule states that the derivative is \( nt^{n-1} \).
This means you multiply by the exponent and subtract 1 from the exponent to find the derivative. It's simple and efficient.

To apply this, let's consider an example: the term \( 2t^{3/2} \). First, identify the power of \( t \), which is \( 3/2 \). According to the power rule, the derivative of \( t^{3/2} \) is \( \frac{3}{2}t^{1/2} \).
Then, multiply this by the coefficient of the term, which is 2 in this case, resulting in \( 3t^{1/2} \).

• Multiply the original exponent with the coefficient.
• Subtract one from the original exponent to get the new power.
• Simplify if necessary.

This rule is crucial for efficiently handling polynomial expressions in calculus, where identifying the exponent and applying the rule allows you to compute derivatives in seconds.

What is Differentiation?

Differentiation is a fundamental concept in calculus that deals with rates of change. It's the process of finding a derivative, which measures how a function changes as its input changes. In essence, it's about understanding how one quantity varies concerning another.

This process is crucial for analyzing curves and modeling real-world behaviors, like velocity in physics or maximizing profit in business. Differentiation transforms a static function into a dynamic tool that tells us how rapidly and in what direction changes occur.

To differentiate a function, identify each term and apply relevant rules like the power rule for polynomials or the chain rule for composite functions. For example, differentiating \( s = 2t^{3/2} + 3e^2 \), involves

• Differentiating \( 2t^{3/2} \) using the power rule as described earlier.
• Recognizing constants like \( 3e^2 \) become zero because they don't change with \( t \).
• Combining the derivatives to get the overall derivative of the function.

In practice, differentiation provides valuable insights into the behavior and properties of functions across various fields.

The Role of the Natural Logarithm in Calculus

Natural logarithms play a significant role in calculus, often appearing in growth processes and in simplifying mathematical expressions. The base of the natural logarithm is the constant \( e \), where \( e \approx 2.71828 \). It's a unique constant because the derivative of \( e^x \) is itself, \( e^x \), offering simplicity in calculus operations.

When we encounter terms like \( e^2 \) in differentiation, it's crucial to recognize them as constants because \( 2 \) is a fixed exponent. These terms, however, become interesting when the exponent is variable, such as \( e^t \), where differentiation involves applying the chain rule.

The use of natural logarithms extends beyond differentiation to integrals, where they help simplify the integration of rational functions, and in Euler's formula, which elegantly links trigonometric functions with exponential growth:

• Natural logarithms transform multiplicative processes into additive ones, simplifying complex expressions.
• They often appear in continuous growth models like population dynamics or compound interest.
• Their properties are widely used in mathematical proofs and solving equations.

Understanding natural logarithms allows us to handle exponential phenomena in a structured, mathematically sound way, bridging the gap between algebraic expressions and applicable calculus principles.