Problem 47 - Applied Calculus

Question

If \(h=f^{n}\), show that $$ \frac{\left(f^{n}\right)^{\prime}}{f^{n}}=n \frac{f^{\prime}}{f} $$

Step-by-Step Solution

Answer

The identity \( \frac{(f^n)'}{f^n} = n \frac{f'}{f} \) is proven by using the power rule for derivatives.

Step 1: Understand the Problem

We need to show that when \( h = f^{n} \), then \( \frac{\left(f^{n}\right)^{\prime}}{f^{n}}=n \frac{f^{\prime}}{f} \). This means finding the derivative of \( f^{n} \) with respect to \( f \), and then dividing it by \( f^{n} \).

Step 2: Take the Derivative of \( f^n \)

To differentiate \( f^n \), we use the power rule for differentiation: if \( u = f(x) \) and \( h(u) = u^n \), then \( h'(u) = n u^{n-1} \cdot u' \). Applying this, we have:\[ (f^{n})' = n f^{n-1} \cdot f' \]

Step 3: Express the Derivative in Terms of \( f^n \)

Now, replace \( f^{n-1} \) in the derivative expression. We have:\[ (f^{n})' = n f^{n-1} \cdot f' = n \cdot \frac{f^n}{f} \cdot f' \]

Step 4: Simplify the Expression

Next, divide the result by \( f^n \):\[ \frac{(f^{n})'}{f^{n}} = \frac{n \cdot f^{n-1} \cdot f'}{f^{n}} = n \frac{f^{n-1} \cdot f'}{f^{n}} = n \frac{f'}{f} \]

Step 5: Conclude the Identity

After simplifying, we confirm that the identity holds:\[ \frac{(f^{n})'}{f^{n}} = n \frac{f'}{f} \]Thus, the expression \( \frac{(f^n)'}{f^n} = n \frac{f'}{f} \) is proven.

Key Concepts

Power rule

The Power Rule is a fundamental concept in calculus. It's incredibly useful when you need to differentiate power functions, which are functions of the form \( f(x) = x^n \). The rule states that if you have a function \( u(x) = x^n \), then its derivative \( u'(x) \) is given by multiplying the exponent \( n \) by \( x^{n-1} \). In a formula, it looks like this:

• \( u'(x) = n \cdot x^{n-1} \)

This approach simplifies finding the derivative of power functions because you only need to lower the exponent by one and multiply by the original exponent. The Power Rule is the backbone of taking derivatives quickly and efficiently in calculus.
When you apply the Power Rule in exercises, always check the function's form. It must be written as a power of \( x \). If it is a more complex function, you might need additional rules like the Chain Rule to help you.

Chain rule

The Chain Rule is another pivotal part of differentiation in calculus. It's particularly important when dealing with composite functions, where one function is nested inside another. A composite function might look like \( h(x) = f(g(x)) \), meaning that \( h(x) \) is a function of \( g(x) \), which is itself a function of \( x \).
To differentiate such a function, we express the Chain Rule as:

• \( h'(x) = f'(g(x)) \cdot g'(x) \)

Essentially, you take the derivative of the outer function \( f \) with respect to its variable, then multiply it by the derivative of the inner function \( g \). This process allows you to "chain" together derivatives, hence the name "Chain Rule."
It's crucial for tackling more complex derivatives involving multiple layers of functions. Whenever you spot a composition of functions, think of the Chain Rule as your go-to method.

Differentiation

Differentiation is the process of finding the derivative of a function. A derivative, at its core, measures how a function changes as its input changes. It's akin to finding the slope of a function at a specific point on a curve.

The main goal of differentiation is to determine the rate of change of a function's value concerning its input variable. In calculus, derivatives provide insights into the behavior of functions, such as where they increase or decrease and their overall shape.

• The notation for derivatives includes \( f'(x) \), \( \frac{df}{dx} \), or \( Df \).

The act of differentiation involves applying rules such as the Power Rule, Chain Rule, and others. These rules simplify obtaining derivatives from functions of varying complexities.
Through differentiation, you can systematically find tangent lines, solve optimization problems, and even find solutions to differential equations.

Exponentiation

Exponentiation is the process of raising a number or expression to a power. In mathematics, it's written as \( a^b \), where \( a \) is the base and \( b \) is the exponent. This operation is foundational in mathematics and appears frequently in calculus.

When speaking of differentiating exponential functions, the rules become slightly more nuanced. For instance, when you differentiate \( f(x) = x^n \), you apply the Power Rule, and when differentiating functions like \( f(x) = a^x \), you apply different techniques suitable for exponential growth and decay functions.

• In calculus, exponentiation is manipulated using logarithms for simplicity while differentiating.

Understanding exponentiation is pivotal, as many functions encountered in real-life scenarios involve exponential growth or decay. It relates closely to differentiation since rates of change often involve exponential relationships.
Knowing how exponentiation integrates with differentiation allows you to solve many calculus problems, particularly those dealing with growth rates, such as compounding interest or population growth.