Problem 4 - Calculus Early Transcendentals

Question

Let \(f(x)=5 \sqrt{x}\) and \(g(x)=4+\cos x\) (a) Find \((f \circ g)(x)\) and \((f \circ g)^{\prime}(x)\) (b) Find \((g \circ f)(x)\) and \((g \circ f)^{\prime}(x)\)

Step-by-Step Solution

Answer

(a) \((f \circ g)(x) = 5\sqrt{4 + \cos x}\), \((f \circ g)'(x) = -\frac{5 \sin x}{2\sqrt{4+\cos x}}\). (b) \((g \circ f)(x) = 4 + \cos(5\sqrt{x})\), \((g \circ f)'(x) = -\frac{5 \sin(5\sqrt{x})}{2\sqrt{x}}\).

Step 1: Find \( (f \circ g)(x) \)

The composition \((f \circ g)(x)\) means substituting \(g(x)\) into \(f(x)\). Substitute \(g(x) = 4 + \cos x\) into \(f(x) = 5 \sqrt{x}\). So, \((f \circ g)(x) = 5 \sqrt{4 + \cos x}\).

Step 2: Differentiate \( (f \circ g)(x) \)

To find \((f \circ g)^{\prime}(x)\), use the chain rule, which is \((f(g(x)))' = f'(g(x)) \cdot g'(x)\). Here, \(f'(x) = \frac{5}{2\sqrt{x}}\) and \(g'(x) = -\sin x\). Therefore, \((f \circ g)^{\prime}(x) = \frac{5}{2\sqrt{4 + \cos x}} \cdot (-\sin x) = -\frac{5 \sin x}{2 \sqrt{4 + \cos x}}\).

Step 3: Find \( (g \circ f)(x) \)

The composition \((g \circ f)(x)\) involves substituting \(f(x)\) into \(g(x)\). Substitute \(f(x) = 5 \sqrt{x}\) into \(g(x) = 4 + \cos x\) to get \((g \circ f)(x) = 4 + \cos(5 \sqrt{x})\).

Step 4: Differentiate \( (g \circ f)(x) \)

Use the chain rule again to differentiate \((g \circ f)(x) = 4 + \cos(5\sqrt{x})\). The derivative is \((g \circ f)^{\prime}(x) = -\sin(5\sqrt{x}) \cdot \frac{5}{2\sqrt{x}} = -\frac{5 \sin(5\sqrt{x})}{2\sqrt{x}}\).

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus used to differentiate composite functions. When you have two functions nested inside one another, this rule helps you find the derivative easily. Formally, the chain rule states that if you have a function composed as \( f(g(x)) \), its derivative is \( f'(g(x)) \cdot g'(x) \).

Here's how it works:

• First, determine the derivative of the outer function, \( f \), while treating the inner function \( g(x) \) as a constant.
• Next, find the derivative of the inner function, \( g(x) \).
• Multiply these two results to get the overall derivative.

Using the chain rule becomes crucial when you want to find the derivative of something like \( (f \circ g)'(x) \). It's like peeling away layers of an onion, starting from the outermost layer.

Derivatives of Functions

Derivatives represent the rate of change of a function with respect to its variable. This concept is central to calculus and essential in various fields such as physics, engineering, and economics.

To find a derivative, you need to understand basic formulas and rules, like the power rule \( \frac{d}{dx} x^n = nx^{n-1} \), and the product, quotient, and chain rules.

• The power rule helps when you deal with variables raised to a power.
• Understanding how each part of a combined function contributes to changes in its overall form will lead to precise calculations of derivatives.
• Practice with these rules makes finding the slopes of tangents much easier and applications like velocity or acceleration more understandable.

Derivatives allow you to find not just slopes but also how a function behaves, identifying maxima and minima through critical points.

Composite Functions

Composite functions are created when one function is applied to the result of another function. Written as \( (f \circ g)(x) = f(g(x)) \), they offer a way to combine simple functions to create more complex ones.

Understanding composite functions involves:

• Recognizing which function is being applied first and which second.
• Plugging the output of the inner function \( g(x) \) into the outer function \( f \).
• Ensuring you correctly apply transformations that the inner function introduces before applying the outer function's rules.

For example, given \( f(x)=5 \sqrt{x} \) and \( g(x)=4 + \cos x \), to compose \( f \) after \( g \), you replace \( x \) in \( f(x) \) with \( g(x) \), giving \( (f \circ g)(x) = 5 \sqrt{4 + \cos x} \). This process creates a bridge between different mathematical expressions, making it easier to analyze their combined behavior.