Problem 38 - Calculus Single Variable

Question

Calculate the derivative of the given xpression with respect to \(x\). $$ \sin \left(\pi^{x}\right) $$

Step-by-Step Solution

Answer

The derivative is \( \pi^x \ln(\pi) \cos(\pi^x) \).

Step 1: Identify the Composition

The given expression is \( \sin \left( \pi^x \right) \). It is composed of the sine function and the exponential function. We can identify it as a composition of the outer function \( f(u) = \sin(u) \) and the inner function \( u(x) = \pi^x \).

Step 2: Differentiate the Outer Function

Differentiate the outer function \( f(u) = \sin(u) \) with respect to \( u \). The derivative of \( \sin(u) \) is \( \cos(u) \). So we have \( f'(u) = \cos(u) \).

Step 3: Differentiate the Inner Function

Differentiate the inner function \( u(x) = \pi^x \) with respect to \( x \). The derivative is obtained using the exponential differentiation formula: \( \frac{d}{dx}(a^x) = a^x \ln(a) \). Thus, \( \frac{d}{dx}(\pi^x) = \pi^x \ln(\pi) \).

Step 4: Apply the Chain Rule

Combine the results using the chain rule. The derivative of the composite function \( \sin(\pi^x) \) is \( f'(u) \cdot u'(x) \). Therefore, we have \( \frac{d}{dx}(\sin(\pi^x)) = \cos(\pi^x) \cdot (\pi^x \ln(\pi)) \).

Step 5: Express in Simplified Form

Multiply the expressions from Step 4 to get the final derivative. Thus, the derivative is \( \pi^x \ln(\pi) \cos(\pi^x) \).

Key Concepts

Chain Rule

The chain rule is an essential technique in calculus used to find the derivative of composite functions. A composite function is one where one function is nested inside another, like the \(\sin(\pi^x)\) in our exercise. The chain rule allows us to differentiate such functions efficiently.

To use the chain rule, follow these steps:

• Identify the inner and the outer functions. In our expression, the inner function is \(u(x) = \pi^x\) and the outer function is \(f(u) = \sin(u)\).
• Differentiate the outer function with respect to its variable. This means calculating \(f'(u)\) which for \(\sin(u)\) is \(\cos(u)\).
• Next, differentiate the inner function with respect to \(x\). Thus, the derivative of \(u(x) = \pi^x\) is \(\pi^x \ln(\pi)\).
• Finally, combine these derivatives using the chain rule formula: \(f'(u) \cdot u'(x)\). This means multiplying \(\cos(\pi^x)\) by \(\pi^x \ln(\pi)\).

Each step of this process must be followed to correctly find the derivative of composite functions. The chain rule helps us understand how functions interact and change when combined.

Exponential Functions

Exponential functions are a class of functions where a constant base is raised to a variable exponent, usually expressed as \(a^x\). In our problem, the base is \(\pi\) and the exponent is \(x\), giving us the function \(\pi^x\).

Exponential functions have unique properties:

• They grow (or decay) very quickly, depending on whether the base is greater or less than one.
• Their derivatives record this rapid rate of change.
• For derivative purposes, the formula \(\frac{d}{dx}(a^x) = a^x \ln(a)\) is paramount. This expression captures the essence of how exponential functions change.
• In our specific exercise, the derivative of \(\pi^x\) involves \(\ln(\pi)\), which accounts for the base \(\pi\). This natural logarithm of the base scales the rate of change of the original function.

Understanding exponential functions is vital as they appear frequently in mathematics, modeling real-world phenomena like population growth and radioactive decay.

Trigonometric Derivatives

Trigonometric functions, like sine and cosine, are foundational in calculus for modeling periodic phenomena. In our expression \(\sin(\pi^x)\), we deal with the sine function. The sine, being a smooth periodic function, has a well-known derivative:

• The derivative of \(\sin(u)\) with respect to \(u\) is \(\cos(u)\). This relationship reflects how the slope of the sine wave changes as it cycles around.
• When paired with the chain rule, this particular derivative helps us manage complex expressions like \(\sin(\pi^x)\).

By understanding how trig functions behave and how their derivatives help determine rates of change, we gain insight into predicting and analyzing wave patterns, oscillations, and other phenomena. In applying these derivatives, the composite nature of the function is revealed, such as when sine interacts with the exponential nature of \(\pi^x\). Consequently, using these derivatives helps simplify complex mathematical problems into manageable steps.