Problem 19 - Calculus of a Single Variable

Question

Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ f(t)=\left(\frac{1}{t-3}\right)^{2} $$

Step-by-Step Solution

Answer

The derivative of the given function is \(f'(t)=\frac{-2}{(t-3)^3}\).

Step 1: Rewrite the Function

Rewrite the function as \(f(t)=(t-3)^{-2}\). This form is easier to differentiate as it's a power function.

Step 2: Apply the Chain Rule

The chain rule states that the derivative of a composite function is derived by replacing the inner function with it's derivative. So, apply the chain rule and get the derivative as \(f'(t)=-2(t-3)^{-3}\times 1\).

Step 3: Simplify the Expression

Simplify the expression by bringing \(t-3\) back to the denominator. So, \(f'(t)=\frac{-2}{(t-3)^3}\).

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus used to find the derivative of a composite function. It's crucial when functions are nested within each other, like in our exercise with the function \(f(t)=\left(\frac{1}{t-3}\right)^{2}\). The chain rule formula is:

• If you have a function \(y=f(u)\) and \(u=g(x)\), then the derivative \(\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}\).

This means you find the derivative of the outer function, then multiply it by the derivative of the inner function. In our exercise, \(f(t)\) can be seen as an outer function \(u^2\), where \(u=\frac{1}{t-3}\). So, the chain rule helps in piecing these derivatives together effectively. Remember:

• Identify the inner and outer functions.
• Differentiate them separately.
• Multiply the results together.

Power Function

A power function has the form \(x^n\), where \(x\) is a variable and \(n\) is a constant exponent. The key to differentiating power functions is applying the power rule, which states:

• \(\frac{d}{dx} x^n = n \cdot x^{n-1}\).

In the context of our exercise, we first rewrite \(\left(\frac{1}{t-3}\right)^2\) as \((t-3)^{-2}\). This transformation into a power function makes it much easier to differentiate. Why is this helpful? Because:

• The power rule directly applies, simplifying the process.
• The exponent drops in front as a coefficient, and the new exponent is reduced by one.

By doing so, the problem of finding the derivative becomes more straightforward using a straightforward method.

Composite Function

A composite function is formed when one function is applied to the results of another, such as \(f(g(x))\). In our specific case, \(f(t) = \left(\frac{1}{t-3}\right)^2\), it consists of a function inside another. Recognizing this nature of the function is vital, as it indicates the need to use the chain rule. Here's what to consider:

• The outer function could be \(u^2\).
• The inner function is \(u = \frac{1}{t-3}\).

In practical terms, whenever you see a function within another, think about the chain rule. Composite functions allow us to dissect and handle complex derivatives by separating tasks into manageable steps. Once you break down the functions, applying rules like the power rule makes the differentiation a simple arithmetic sequence.