Question

Derivatives of tower functions (or \(g^{h}\) ) Find the derivative of each function and evaluate the derivative at the given value of \(a\). $$f(x)=(4 \sin x+2)^{\cos x} ; a=\pi$$

Short answer

Answer: The derivative of the function \(f(x)\) evaluated at \(x=\pi\) is equal to \(\boxed{2}\).

Step-by-step solution

Recognize the composition of functions

We can write the given function \(f(x)\) as a composition of two functions: \(f(x)=g^{h}\), where \(g(x)=4\sin{x}+2\) and \(h(x)=\cos{x}\). Our goal is to find the derivative \(f'(x)\) and evaluate it at \(x=\pi\).

Apply the chain rule

To find the derivative \(f'(x)\), we need to apply the chain rule, which states that if \(y=g^h\), then its derivative is given by: $$\frac{dy}{dx} = (g^h)[(\ln g)\frac{dh}{dx} + \frac{h}{g}\frac{dg}{dx}]$$

Differentiate \(g(x)\) and \(h(x)\)

Now we need to find the derivatives of \(g(x)\) and \(h(x)\). Derivative of \(g(x)\): $$\frac{dg}{dx} = \frac{d}{dx} (4\sin{x} + 2) = 4\cos{x}$$ Derivative of \(h(x)\): $$\frac{dh}{dx} = \frac{d}{dx} (\cos{x}) = -\sin{x}$$

Substitute the expressions into the chain rule formula

Now we have everything we need to determine \(f'(x)\). Substitute the expressions for \(g(x)\), \(h(x)\), \(\frac{dg}{dx}\), and \(\frac{dh}{dx}\) into the chain rule formula: $$f'(x) = (g^h)[(\ln g)\frac{dh}{dx} + \frac{h}{g}\frac{dg}{dx}]$$ $$f'(x) = ((4 \sin{x}+2)^{\cos{x}})[(\ln (4 \sin{x}+2))(-\sin{x}) + \frac{\cos{x}}{(4 \sin{x}+2)}(4\cos{x} )]$$

Evaluate the derivative at \(x=\pi\)

Finally, we can substitute \(x=\pi\) into our expression for \(f'(x)\): $$f'(\pi) = ((4 \sin{\pi}+2)^{\cos{\pi}})[(\ln (4 \sin{\pi}+2))(-\sin{\pi}) + \frac{\cos{\pi}}{(4 \sin{\pi}+2)}(4\cos{\pi} )]$$ Since \(\sin{\pi}=0\), \(\cos{\pi}=-1\), the expression becomes: $$f'(\pi) = ((4(0)+2)^{-1})[(\ln (2))(0) + \frac{-1}{2}(4(-1))]$$ Simplify to get: $$f'(\pi) = 2^{-1}(4) = \boxed{2}$$ Thus, the derivative of the function \(f(x)\) evaluated at \(x=\pi\) is equal to 2.

Key concepts

Derivatives of Tower Functions

Tower functions are fascinating because they involve an exponential function raised to another function. This results in a composite structure that requires special techniques for differentiation. In such scenarios, we handle the function as if it follows the form of one function raised to another, written as \(g^h\), where both \(g(x)\) and \(h(x)\) are themselves functions of \(x\). This conjugation of functions makes differentiation more intricate but also more insightful as it allows us to explore how steeply and dynamically functions can change. For example, in the function \(f(x)=(4 \sin x + 2)^{\cos x}\), we have \(g(x) = 4 \sin x + 2 \) as the base and \(h(x) = \cos x\) as the exponent. Recognizing this structure is the first step in identifying how to apply calculus methods like the chain rule for differentiation.

Chain Rule

The chain rule is critical when differentiating composite functions, especially those involved in tower functions. In simple terms, the chain rule states that to differentiate a composite function \(y = g(h(x))\), we multiply the derivative of the outer function \(g\) evaluated at \(h(x)\) by the derivative of the inner function \(h(x)\). When dealing with tower functions, the formula adapts to account for the derivative of the form \(g^h\). The chain rule for this form becomes:

• \((g^h) \left[ (\ln g) \frac{dh}{dx} + \frac{h}{g} \frac{dg}{dx} \right]\)

This formula allows us to account for changes in both the base and the exponent, offering a comprehensive way to manage the intricacies of tower functions in calculus. By systematically applying this rule, we can decompose and tackle the differentiation of functions with multiple layers of complexity.

Differentiation

Differentiation is the process through which we find the rate at which a function is changing at any given point. It's a foundational concept in calculus, providing valuable information about the nature and behavior of functions. When differentiating a complex function, breaking it into primary functions or elements helps. In our exercise:

• First, differentiate \(g(x) = 4 \sin x + 2\) to get \( \frac{dg}{dx} = 4 \cos x\).
• Then differentiate \(h(x) = \cos x\) to find \( \frac{dh}{dx} = - \sin x\).

Putting together these derivatives provides the tools needed to apply the chain rule effectively. Differentiation essentially uncovers the slope of the function at every point, thus serving as a microscope for the deeper analysis of function behavior.

Composite Functions

Composite functions are those where one function is nested inside another. In the context of our task, understand \(f(x)=(4 \sin x + 2)^{\cos x}\) requires identifying both the base \(g(x)\) and the exponent \(h(x)\). These are then combined in a way that forms a single, more complex function. Such compositions are prevalent in calculus and serve purposes like modeling multi-scenario situations in real-world problems. Solving for derivatives of composite functions typically demands breaking them down using the rules of derivatives like the chain rule, as seen in this exercise. With \(f(x)\), we appreciate the different roles \(g(x)\) and \(h(x)\) play, especially as they intertwine to impact overall behavior. Recognizing these layers is essential for effective differentiation and further calculus operations.