Problem 26 - Applied Calculus

Question

Find the derivative. Assume \(a, b, c, k\) are constants. $$h(t)=\frac{3}{t}+\frac{4}{t^{2}}$$

Step-by-Step Solution

Answer

The derivative is \( h'(t) = \frac{-3}{t^2} - \frac{8}{t^3} \).

Step 1: Rewrite the Function

The given function is \( h(t) = \frac{3}{t} + \frac{4}{t^2} \). First, rewrite these terms using exponents. Remember that \( \frac{1}{t^n} = t^{-n} \). Therefore, \( \frac{3}{t} = 3t^{-1} \) and \( \frac{4}{t^2} = 4t^{-2} \). So, the function becomes \( h(t) = 3t^{-1} + 4t^{-2} \).

Step 2: Differentiate the Function

To differentiate each term individually, we apply the power rule which states \( \frac{d}{dt}[t^n] = nt^{n-1} \). For the first term, \( \frac{d}{dt}[3t^{-1}] = 3(-1)t^{-2} = -3t^{-2} \). For the second term, \( \frac{d}{dt}[4t^{-2}] = 4(-2)t^{-3} = -8t^{-3} \).

Step 3: Combine the Derivatives

Combine the derivatives from the previous step to obtain the derivative of the original function. The derivative of \( h(t) \) becomes \( h'(t) = -3t^{-2} - 8t^{-3} \).

Step 4: Simplify the Derivative

To simplify the final result, you can write each term with positive exponents: \( -3t^{-2} = \frac{-3}{t^2} \) and \( -8t^{-3} = \frac{-8}{t^3} \). Thus, the simplified derivative is \( h'(t) = \frac{-3}{t^2} - \frac{8}{t^3} \).

Key Concepts

Power Rule

In calculus, the power rule is a basic principle used to find the derivative of functions with exponents. The power rule states that if you have a function in the form of \( t^n \), the derivative is obtained by multiplying the exponent \( n \) by the variable \( t \) raised to the power of \( n-1 \).
For example, the derivative of \( t^3 \) is \( 3t^2 \). This rule simplifies the process of differentiation.

When applying the power rule:

• Identify functions of the form \( t^n \).
• Multiply by the original exponent.
• Decrease the exponent by one.

This rule is particularly useful when dealing with polynomial functions, which are sums of power terms.

Differentiation of Polynomials

Polynomials are expressions consisting of variables with non-negative integer exponents. Differentiating polynomials involves applying the power rule to each term individually.
For example, for a polynomial like \( h(t) = 3t^2 - 2t + 5 \), you would differentiate to get \( h'(t) = 6t - 2 \).

Steps to differentiate polynomials:

• Apply the power rule to each term separately.
• Constants (like 5) have a derivative of zero since they don't change with the variable.
• Sum all derivative terms.

The process is straightforward as it allows us to easily handle functions with multiple terms.

Exponent Rules

Exponent rules are the mathematical guidelines for manipulating expressions with exponents. They help in rewriting expressions, which is essential before applying the power rule in differentiation.
Key exponent rules that apply here include:

• \( t^{-n} = \frac{1}{t^n} \): Negative exponents turn into fractions.
• \( t^m \times t^n = t^{m+n} \): Multiply and add exponents.
• \( \frac{t^m}{t^n} = t^{m-n} \): Subtract exponents in division.

Understanding these rules makes handling exponents in functions and calculus much more manageable.

Chain Rule

The chain rule is used when differentiating composite functions, or functions within another function. It combines derivatives of inside and outside functions.
Although not directly applied in the given problem, understanding the chain rule is crucial for more complex derivatives.

The formula for the chain rule is:
If \( y = f(g(t)) \), then \( \frac{dy}{dt} = f'(g(t)) \cdot g'(t) \).

Steps involved:

• Differentially treat each part of the function.
• Multiply the derivative of the outer function by the derivative of the inner one.
• Chain rules are essential in integration too.

This rule is vital in calculus, offering a method to handle nested functions effectively.