Problem 23 - Applied Calculus

Question

Find the derivative. Assume \(a, b, c, k\) are constants. $$y=3 t^{5}-5 \sqrt{t}+\frac{7}{t}$$

Step-by-Step Solution

Answer

The derivative is \( y' = 15t^4 - \frac{5}{2}t^{-1/2} - 7t^{-2} \).

Step 1: Identify the Terms

The given function is \( y = 3t^5 - 5\sqrt{t} + \frac{7}{t} \). Here, \( 3t^5 \) is a power of \( t \), \( 5\sqrt{t} \) can be rewritten as \( 5t^{1/2} \), and \( \frac{7}{t} \) can be rewritten as \( 7t^{-1} \).

Step 2: Differentiate Each Term Separately

Differentiate each term using the power rule. The derivative of \( 3t^5 \) is \( 15t^4 \), the derivative of \( -5t^{1/2} \) is \( -\frac{5}{2}t^{-1/2} \), and the derivative of \( 7t^{-1} \) is \( -7t^{-2} \).

Step 3: Combine the Derivatives

Combine all the derivatives from Step 2 to form the derivative of the entire function. Thus, the derivative \( y' \) is \( 15t^4 - \frac{5}{2}t^{-1/2} - 7t^{-2} \).

Key Concepts

Power Rule

The power rule is a fundamental tool in calculus used for finding derivatives. If you have a function in the form of a power of a variable, such as \( t^n \), the power rule states that the derivative is found by multiplying the exponent by the coefficient, and then reducing the exponent by one:

• If \( f(t) = t^n \), then \( f'(t) = n \cdot t^{n-1} \).

In the given exercise, we apply the power rule to each term individually. For the term \( 3t^5 \), applying the power rule gives \( 15t^4 \). Similarly, for \( 5t^{1/2} \), we rewrite \( \sqrt{t} \) as \( t^{1/2} \) and then derive it to get \( \frac{1}{2} \times 5 \times t^{-1/2} \), which simplifies to \(-\frac{5}{2}t^{-1/2} \). For the term \( \frac{7}{t} \), rewrite it as \( 7t^{-1} \), and applying the power rule results in \(-7t^{-2} \).

The power rule is straightforward, which makes it a go-to method in calculus when dealing with powers of a variable.

Differentiation Techniques

Differentiation is the process of finding the derivative of a function. It is used to determine how a function changes as its input changes. One of the most common differentiation techniques is the power rule, but there are several others that are used depending on the form of the function.

Some other useful differentiation techniques include:

• Product Rule: Used when differentiating functions that are products of two or more functions.
• Quotient Rule: Used for finding the derivative of a quotient of two functions.
• Chain Rule: Used for differentiating composite functions.

In this exercise, we exclusively use the power rule since all terms can be simplified to power functions before differentiating. By individually differentiating each term, we allow ourselves to treat each term independently, making complex functions simpler to handle. It's often easier to break complex functions down, apply the most suitable differentiation technique to each part, and then combine the results to find the overall derivative.

Function Transformation

Function transformation involves rewriting functions to enable easier manipulation and solution, especially when applying calculus techniques such as differentiation.

In this problem, several transformations are essential to facilitate the differentiation process:

• Rewrite roots as fractional exponents: \( \sqrt{t} \) becomes \( t^{1/2} \).
• Rewrite reciprocal terms as negative exponents: \( \frac{7}{t} \) becomes \( 7t^{-1} \).

These transformations simplify using the power rule directly as no additional complicated operations, such as division or root differentiation, are involved. Transforming functions not only eases the computation but also provides clarity regarding the terms and their respective derivatives. Understanding function transformation is a crucial skill in calculus, aiding in many different types of problems beyond just differentiation.