Problem 29 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(t)=\frac{4}{t^{4}}-\frac{3}{t^{3}}+\frac{2}{t}\)

Step-by-Step Solution

Answer

The derivative of the function \(f(t)=\frac{4}{t^{4}}-\frac{3}{t^{3}}+\frac{2}{t}\) is \(f'(t) = \frac{-16}{t^5} + \frac{9}{t^4} - \frac{2}{t^2}\).

Step 1: Rewrite the function in a more workable form

First, let's rewrite the function as the sum of three terms with exponents: \(f(t) = 4t^{-4} - 3t^{-3} + 2t^{-1}\)

Step 2: Apply the power rule to each term

Now, let's apply the power rule to each term: \(f'(t) = -4(4)t^{-4-1} - 3(-3)t^{-3-1} + 2(-1)t^{-1-1}\)

Step 3: Simplify the expression

Finally, let's simplify the expression: \(f'(t) = -16t^{-5} + 9t^{-4} - 2t^{-2}\)

Step 4: Rewrite the function in its original form

Let's rewrite the function back in the original form using fractions: \(f'(t) = \frac{-16}{t^5} + \frac{9}{t^4} - \frac{2}{t^2}\) So the derivative of the given function is: \(f'(t) = \frac{-16}{t^5} + \frac{9}{t^4} - \frac{2}{t^2}\)

Key Concepts

Power Rule

One of the most ubiquitous tools in calculus is the power rule for differentiation. At its core, the power rule is a quick and straightforward method for finding the derivative of a function where the variable has an exponent. This rule states that if you have a function of the form \( f(x) = ax^n \), where \( a \) is a constant and \( n \) is a real number, the derivative of that function, \( f'(x) \), is given by multiplying the exponent by the coefficient and then reducing the exponent by one. In mathematical terms, \( f'(x) = anx^{n-1} \).

For example, let's say we have a term \( 4x^3 \). According to the power rule, the derivative would be \( 4 \times 3x^{3-1} = 12x^2 \). This rule is invaluable because it simplifies the process of differentiation, allowing for the quick calculation of derivatives, particularly in polynomial functions. In the case of the example exercise, each term of the function \( f(t) = 4t^{-4} - 3t^{-3} + 2t^{-1} \) was differentiated using the power rule resulting in the derivative \( f'(t) \).

Derivative of a Function

The derivative of a function at a certain point essentially represents the rate at which the function's value is changing at that point. In other words, it gives us the slope of the tangent line to the function's graph at a given value of \( x \). This can interpret how fast or slow the function is increasing or decreasing, providing insight into the function's behavior.

In practice, finding the derivative involves applying rules like the power rule, product rule, quotient rule, or chain rule, depending on the function's complexity. Once we have the derivative, \( f'(x) \), we can evaluate it at different points to understand how the function behaves locally. Notably, if the derivative is positive, the function is increasing, and if it is negative, the function is decreasing. Zeroes of the derivative can indicate critical points, where the function may have local maxima, minima, or inflection points.

Simplifying Expressions

Simplifying expressions in mathematics is akin to tidying up a room—the goal is to make it orderly and more manageable. When we simplify an expression, we reduce it to its most basic form, making it easier to understand and work with. This often involves combining like terms, canceling out factors, and applying algebraic rules to achieve a neater, more concise expression.

In the context of calculus and differentiation, after taking the derivative of a function, it is common practice to simplify the result. This often means getting rid of negative exponents, as in the exercise example where \( f'(t) = -16t^{-5} + 9t^{-4} - 2t^{-2} \) was presented in a cleaner, more conventional fraction form: \( f'(t) = \frac{-16}{t^5} + \frac{9}{t^4} - \frac{2}{t^2} \). Simplification is not just an aesthetic choice—it can make further calculations much easier and help to avoid mistakes in complex, multi-step problems.

The process of simplifying is not just about appearance; it's often necessary for the subsequent application of the function. For instance, graphing the function or finding the limits and integrals becomes less prone to error when the expressions are simplified.