Problem 17 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the derivative of each function. \(f(t)=\frac{1}{\sqrt{2 t-3}}\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Inner Function

First, let's identify the function inside the square root: The inner function is \(u(t) = 2t - 3\).

Step 2: Differentiate the Inner Function

Now, we need to differentiate the inner function with respect to t. \[\frac{d}{d t}u(t) = \frac{d}{d t}(2t - 3) = 2\]

Step 3: Apply Chain Rule to the Square Root

Now, let's apply the chain rule to the square root function. We have \(y = \sqrt{u}\) and we know \(u(t) = 2t - 3\). So, we want to find \(\frac{dy}{dt}\). Using the chain rule, we have: \[\frac{dy}{dt} = \frac{dy}{du} \cdot \frac{du}{dt}\] Before we proceed, let's find \(\frac{dy}{du}\):

Step 4: Differentiate the Square Root Function

The function y with respect to u is \(y = \sqrt{u}\), or \(y = u^{\frac{1}{2}}\). We differentiate with respect to u: \[\frac{dy}{du} = \frac{1}{2}u^{-\frac{1}{2}}\]

Step 5: Apply Chain Rule to Find the Derivative

Now, we can go back to finding \(\frac{dy}{dt}\), using the chain rule formula: \[\frac{dy}{dt} = \frac{dy}{du} \cdot \frac{du}{dt} = \frac{1}{2}(2t-3)^{-\frac{1}{2}} \cdot 2\]

Step 6: Find Derivative of Reciprocal Function

Recall that the given function is \(f(t) = \frac{1}{\sqrt{2t-3}}\), which can be written as \(f(t) = (\sqrt{2t-3})^{-1}\). So now, we want to find \(\frac{df}{dt}\). Using the chain rule, we have: \[\frac{df}{dt} = \frac{df}{dy} \cdot \frac{dy}{dt}\] First, differentiate the reciprocal function with respect to y: \(f(y) = y^{-1}\) \[\frac{df}{dy} = -1y^{-2}\]

Step 7: Apply Chain Rule to Find the Final Derivative

Finally, we will use the chain rule to find the derivative of the given function: \[\frac{df}{dt} = \frac{df}{dy} \cdot \frac{dy}{dt} = -1(2t-3)^{-1} \cdot \frac{1}{2}(2t-3)^{-\frac{1}{2}} \cdot 2\] Simplify the expression: \[\frac{df}{dt} = \frac{-2}{(2t-3)^{\frac{3}{2}}}\] The derivative of the given function is \(f'(t) = \frac{-2}{(2t-3)^{\frac{3}{2}}}\).

Key Concepts

Derivative calculation

To find the derivative of a function is to essentially find the rate at which the function's value changes as its input changes. If you think about a car's speedometer - that's akin to a derivative, showing how quickly the car's position is changing over time.

When you see a problem asking you to find a derivative, remember that you might need to:

• Identify parts of the function, such as constants and variables.
• Apply rules, like the power rule or product rule, to simplify your work.
• Manage special functions with unique approaches, such as trigonometric functions or logarithms.

In our case, our function involves both a square root and a reciprocal. Both of these require specific attention. We convert the square root to a fractional exponent, as this format is easier to differentiate. We also address the reciprocal function's specific needs, ensuring we're applying the right approaches for each challenge.

Chain rule in differentiation

The chain rule is a powerful tool that allows us to differentiate complex functions. It is especially helpful when dealing with composite functions, where one function is nested inside another.

A classic example of when to use the chain rule is when you have a function within a function, like our inner function wrapped in a square root. Here’s the approach:

• First, identify the "outer" function (the square root, in our example)
• Then identify the "inner" function (for us it's the expression inside the square root)
• Differentiate each part separately: Start with the outer function while keeping the inner function untouched.
• Multiply the derivative of the outer function by the derivative of the inner function.

Through this process, you're essentially peeling back layers, tackling one piece at a time, then combining the results to understand the full rate of change. Our exercise uses the chain rule to tackle not just the square root, but eventually to arrive at the derivative of the entire complex reciprocal expression.

Reciprocal function differentiation

Differentiating reciprocal functions often requires careful handling, as you're dealing with expressions of the form \(f(x) = \frac{1}{g(x)}\).

The reciprocal function can pose unique challenges because it represents division, and division complicates multiplication's simple rules. When you differentiate an inverse or reciprocal, the negative sign must be carefully managed, along with exponent changes.

Here's a simplified way to handle this:

• Express the reciprocal in a power format, like \(g(x)^{-1}\).
• Apply the chain rule and power rule: derivation puts a negative exponent into play.
• Be mindful of the negative sign introduced during the differentiation process.

This technique ensures that you get the right result, avoiding mistakes in handling the division's complexity. Combining this with the chain rule, our final derivative of the given function was found to be\(f'(t) = \frac{-2}{(2t-3)^{\frac{3}{2}}}\), highlighting just how each of these differentiation strategies builds upon the last for accurate results.