Problem 58 - Thomas Calculus in SI Units

Question

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

Step-by-Step Solution

Answer

\(\frac{dy}{ds} = 2^{s^2} \cdot 2s \cdot \ln(2)\)

Step 1: Identify the function

The function given is \(y = 2^{(s^2)}\). We want to find the derivative \(\frac{dy}{ds}\).

Step 2: Recall the chain rule

The chain rule states that if you have a function \(y=f(g(x))\), then its derivative is \(\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\). Here, \(f(u)=2^u\) and \(g(s)=s^2\), so \(u=g(s)\).

Step 3: Find the derivative of the outer function

For the outer function \(y = 2^u\), the derivative with respect to \(u\) is \(\frac{d}{du}(2^u) = 2^u \cdot \ln(2)\).

Step 4: Find the derivative of the inner function

The inner function is \(u = s^2\). Its derivative with respect to \(s\) is \(\frac{du}{ds} = 2s\).

Step 5: Apply the chain rule

Using the chain rule, the derivative of \(y\) with respect to \(s\) is obtained by multiplying the derivative of the outer function by the derivative of the inner function: \(\frac{dy}{ds} = 2^{s^2} \cdot \ln(2) \cdot 2s\).

Step 6: Simplify the expression

Finally, we simplify the expression for the derivative: \(\frac{dy}{ds} = 2^{s^2} \cdot 2s \cdot \ln(2)\).

Key Concepts

Chain Rule

In calculus, the chain rule is a fundamental method for finding the derivative of a composite function. A composite function occurs when one function is nested within another. By using the chain rule, we can effectively break down a complex operation into simpler parts.

To apply the chain rule, assume you have a composite function in the form of \( y = f(g(x)) \). Here, \( f \) is the outer function and \( g \) is the inner function. The chain rule tells us that the derivative of \( y \) with respect to \( x \) is given by:

• First, take the derivative of the outer function \( f \) with respect to \( g(x) \), which is \( f'(g(x)) \).
• Next, take the derivative of the inner function \( g \) with respect to \( x \), which is \( g'(x) \).
• Finally, multiply these two derivatives together to find the result: \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).

For example, to find the derivative of \( y = 2^{(s^2)} \), we identify the outer function as \( 2^u \) and the inner function as \( s^2 \). This means \( f(u) = 2^u \) and \( g(s) = s^2 \). The chain rule then allows us to determine \( \frac{dy}{ds} = 2^{s^2} \cdot \ln(2) \cdot 2s \).

In essence, the chain rule helps us take on challenging differentiations by dividing them into manageable segments.

Exponential Function Differentiation

Exponential functions are a staple in calculus and mathematics in general. They are defined as functions of the form \( a^x \), where \( a \) is a constant. A specific noteworthy case is when the base \( a = e \), the natural exponential function.

Differentiating exponential functions requires a distinct method because their derivatives often reflect the form of the original function.

• For the function \( y = a^x \), where \( a > 0 \) and \( a eq 1 \), the derivative with respect to \( x \) is \( \frac{dy}{dx} = a^x \cdot \ln(a) \).

This rule can be applied to differentiate any exponential function like the one in our problem, \( y = 2^{(s^2)} \). When differentiating this, we treat the exponent as a separate function, which allows us to use the chain rule effectively.

For example, for our outer function \( f(u) = 2^u \), the differentiation step produces \( 2^u \cdot \ln(2) \). This factor, \( \ln(2) \), comes from the rule for differentiating exponential functions, showing how these types of functions bring in unique constants, dictated by their base.

Derivatives

Derivatives are one of the core tools in calculus. They essentially measure how a function changes as its input changes. In mathematical terms, the derivative provides the rate of change or the slope of the function at any given point.

Understanding derivatives can empower you to tackle various mathematical problems, from predicting trends to optimizing processes.

• The derivative of a function \( y = f(x) \) is often denoted by \( \frac{dy}{dx} \) or \( f'(x) \).
• This represents the instantaneous rate of change of \( y \) with respect to \( x \).

In practical applications, recognizing the type of function you're dealing with helps you choose the best technique to find its derivative. Whether it's applying the power rule, chain rule, or any other method, derivatives map the changing landscape of mathematics.

In our example, deriving \( y = 2^{(s^2)} \) involved leveraging exponential and chain rule concepts. The entire process highlights how derivatives form connections and offer insights between different mathematical ideas.