Problem 62 - Thomas Calculus

Question

Find the derivative of \(y\) with respect to the given independent variable. \begin{equation}y=2^{\left(s^{2}\right)}\end{equation}

Step-by-Step Solution

Answer

The derivative is \(2^{s^2} \cdot 2s \cdot \ln(2)\).

Step 1: Identify the function type

In the given function \(y = 2^{s^2}\), we observe that the variable \(s\) is in the exponent. This implies that it is an exponential function with a variable exponent.

Step 2: Use the Chain Rule

To differentiate a function of the form \(y = a^{u(s)}\), where \(a\) is a constant and \(u(s)\) is a function of \(s\), the chain rule is used: \(\frac{dy}{ds} = a^{u(s)} \cdot \ln(a) \cdot \frac{du}{ds}\).

Step 3: Differentiate the exponent

The exponent \(u(s) = s^2\) is a simple polynomial function. Calculate its derivative: \(\frac{du}{ds} = \frac{d}{ds}(s^2) = 2s\).

Step 4: Apply the Chain Rule formula

Apply the chain rule to find the derivative: \[ \frac{dy}{ds} = 2^{s^2} \cdot \ln(2) \cdot 2s \].

Step 5: Simplify the expression

Combine the constants and simplify the expression: \[ \frac{dy}{ds} = 2^{s^2} \cdot 2s \cdot \ln(2) \].

Key Concepts

Exponential Functions

Exponential functions are fascinating and powerful mathematical expressions where a constant base is raised to a variable exponent. They often take the form \(y = a^x\), where \(a\) is a constant. In our exercise, we have \(y = 2^{s^2}\). Here, 2 is the base, and the exponent is the variable \(s^2\). Exponential functions have a unique property—it is the way they result in rapid growth or decay, depending on the base and the sign of the exponent. When dealing with exponential functions in calculus, we often have to find their derivatives to understand their behavior over different rates of change. An essential point to remember is that the variable in an exponential function's exponent makes differentiation slightly more challenging, which is why specialized rules such as the chain rule are necessary to compute a derivative.

Chain Rule

The chain rule is a super helpful technique in calculus for finding derivatives of composite functions. A composite function is essentially a function within another function. In our case, the function \(y = 2^{s^2}\) is a composite function due to the variable in the exponent.To apply the chain rule, we follow a specific pattern:

• Identify the outer and inner functions. Here, the outer function is the exponential \(2^u\) (where \(u = s^2\)), and the inner function is \(u = s^2\).
• Differentiate the outer function with respect to the inner function. The derivative of \(2^u\) is \(2^u \cdot \ln 2\).
• Differentiate the inner function with respect to its own variable. The derivative of \(s^2\) is \(2s\).

Finally, multiply these derivatives together for the complete derivative by using the chain rule. This powerful approach lets us unravel complexity in differentiating composite functions.

Derivative Calculation

Finding the derivative, especially with exponential functions, requires a systematic approach. In the example \(y = 2^{s^2}\), differentiating involves applying the chain rule.First, we identify that our function, \(y\), can be expressed as \(2^{u(s)}\), where \(u(s) = s^2\). We previously mentioned the derivative of the outer function as \(2^{s^2} \cdot \ln(2)\). Next, we find the derivative of the inner function, \(u(s) = s^2\), treated as a simple polynomial function. This inner function's derivative is \(2s\). Finally, according to the chain rule:

• Multiply the derivative of the outer function by the derivative of the inner function.
• This gives us the complete derivative, which is \(2^{s^2} \cdot 2s \cdot \ln(2)\).

By following these steps, you can confidently tackle differentiation problems involving exponential functions and their composite nature.