Problem 80 - Brief Applied Calculus

Question

Let \(f\) and \(g\) be differentiable functions of \(x\). Assume that denominators are not zero. True or False: \(\frac{d}{d x}\left(\frac{f}{g}\right)=\frac{f^{\prime}}{g^{\prime}}\)

Step-by-Step Solution

Answer

False, the correct derivative uses the quotient rule: \( \frac{f'g - fg'}{g^2} \).

Step 1: Recall the Quotient Rule Formula

The quotient rule for differentiation states that if you have two differentiable functions \( f(x) \) and \( g(x) \), then the derivative of their quotient \( \frac{f}{g} \) is given by \( \left( \frac{f}{g} \right)' = \frac{f'g - fg'}{g^2} \). This formula is different from taking the derivative of each function individually and placing it over the derivative of another.

Step 2: Differentiate \( \frac{f}{g} \) Using the Quotient Rule

Applying the quotient rule to \( \frac{f}{g} \), we find:\[\left( \frac{f}{g} \right)' = \frac{f'g - fg'}{g^2}\]This shows that the derivative is a combination of both \( f' \) and \( g' \) multiplied by the original functions.

Step 3: Compare With Given Expression

The given expression suggests that\[\frac{d}{dx} \left( \frac{f}{g} \right) = \frac{f'}{g'}\]Comparing this with the actual quotient rule gives a different outcome since \( \frac{f'}{g'} \) doesn't include the interaction term \( fg' \), and it has \( g^2 \) in the denominator in the rule.

Step 4: Conclusion

From the comparison, we can conclude that the statement\[\frac{d}{dx} \left( \frac{f}{g} \right) = \frac{f'}{g'}\]is false. The correct derivative involves applying the quotient rule, not simply dividing the derivatives.

Key Concepts

Differentiable Functions

Differentiable functions are an essential concept in calculus, as they are functions that have a derivative at every point in their domain. Being differentiable means that the function is smooth and has no sharp corners or discontinuities where the derivative cannot be defined. For a function to be differentiable, especially in the context of real numbers, two main conditions must be satisfied:

• The function must be continuous at every point in its domain.
• The derivative of the function must exist at each point in its domain.

When dealing with differentiable functions like \( f(x) \) and \( g(x) \), their behavior and properties are essential for calculating derivatives, such as in our exercise scenario. Since the exercise assures us that both \( f \) and \( g \) are differentiable (and we assume \( g(x) \) is never zero), this guarantees that we can safely apply the quotient rule for differentiation.

Derivative

In calculus, the derivative of a function measures how the function's output value changes as its input value changes. It represents the function's rate of change or slope at a particular point. When calculating derivatives, we use various rules like the power, product, and quotient rules, depending on the type of function. The derivative is denoted using different notations: \( f'(x) \), \( \frac{dy}{dx} \), or \( Df(x) \). Calculating the derivative of function combinations, such as quotients of two functions, involves special formulas to ensure that each component's contribution to change is captured. This is where the quotient rule becomes crucial when finding the derivative of \( \frac{f}{g} \). Understanding the role of derivatives helps us solve real-world problems, including speed, growth rates, and optimization.

Quotient Rule Formula

The quotient rule is a formula used to find the derivative of a quotient of two differentiable functions \( f(x) \) and \( g(x) \). The formula is:\[\left( \frac{f}{g} \right)' = \frac{f'g - fg'}{g^2}\]This indicates that to differentiate a fraction of two functions, you:

• Multiply the derivative of the numerator \( f' \) by the denominator \( g \).
• Subtract the product of the numerator \( f \) and the derivative of the denominator \( g' \).
• Divide everything by the square of the denominator \( g^2 \).

It's crucial to note that the derivative is not simply \( \frac{f'}{g'} \), which the incorrect statement in the exercise suggests. The correct application of the quotient rule ensures that both functions' interactions are adequately captured, especially where the non-linear nature of division of functions is concerned. Remember, \( g(x) \) must not be zero to avoid division by zero, making the formula valid.