Problem 61 - Calculus

Question

Find the derivatives in algebraically.$$g(z)=z^{-2}, \text {find } g^{\prime}(2)$$

Step-by-Step Solution

Answer

The derivative of the function at \( z = 2 \) is \(-\frac{1}{4}\).

Step 1: Write the function

The given function is \( g(z) = z^{-2} \). We are asked to find the derivative, \( g'(z) \), and then evaluate it at \( z = 2 \).

Step 2: Differentiate the function

Use the power rule for differentiation which states if \( f(z) = z^n \), then \( f'(z) = n \cdot z^{n-1} \). Here, \( n = -2 \), so the derivative is \( g'(z) = -2 \cdot z^{-3} \).

Step 3: Evaluate the derivative at z = 2

Substitute \( z = 2 \) into \( g'(z) = -2 \cdot z^{-3} \) to find \( g'(2) \). This gives \( g'(2) = -2 \cdot (2)^{-3} \).

Step 4: Compute \(2^{-3}\)

Calculate \(2^{-3}\), which is \( \frac{1}{2^3} = \frac{1}{8} \).

Step 5: Calculate \(g'(2)\)

Substitute \( 2^{-3} = \frac{1}{8} \) into \( g'(2) = -2 \cdot \frac{1}{8} \). Thus, \( g'(2) = -\frac{2}{8} = -\frac{1}{4} \).

Key Concepts

Power Rule

When it comes to finding the derivative of functions in calculus, one of the most commonly used techniques is the power rule. The power rule simplifies the process of differentiation for polynomial functions of the form \( f(z) = z^n \). The rule states that the derivative, \( f'(z) \), is found by multiplying the power \( n \) by the base \( z \) raised to the power \( n-1 \). In simpler terms:

• If your function is \( z^n \), then its derivative will be \( n \cdot z^{n-1} \).

So for a function like \( g(z) = z^{-2} \), applying the power rule means you take \( n = -2 \) and find the derivative \( g'(z) = -2 \cdot z^{-3} \). This technique drastically reduces the complication often found in differentiation, allowing one to quickly move from function to its derivative.

Differentiation

Differentiation is an essential part of calculus that involves calculating the rate at which a function changes. In simpler terms, it's the process of finding a derivative. This is particularly important when you want to understand how a function behaves at various points. The power rule, as we explored earlier, is a handy tool used in the differentiation process for polynomial functions.In our problem, we differentiated \( g(z) = z^{-2} \) using the power rule. Differentiating step by step:

• Take the exponent \( -2 \) and multiply it by the coefficient of \( z \), here the coefficient is implicitly 1. So, \(-2 \times 1 = -2\).
• Decrease the exponent by 1: \(-2 - 1 = -3\).

This process gives us \( g'(z) = -2 \cdot z^{-3} \). Differentiation through the power rule is effective for functions involving powers of a variable, making it an indispensable part of calculus.

Evaluating Derivatives

After finding the derivative of a function, the next step is often evaluating it at a specific point. This helps to determine the behavior of the function at that particular value. Evaluating the derivative involves substituting a given value into the derivative function and simplifying.For our function \( g(z) = z^{-2} \), we derived \( g'(z) = -2 \cdot z^{-3} \). To evaluate this at \( z = 2 \), plug 2 into the derivative:Substitute: \( g'(2) = -2 \cdot (2)^{-3} \).

• First, calculate \( (2)^{-3} \):
  • This equals \( \frac{1}{2^3} = \frac{1}{8} \).
• Then, multiply: \( g'(2) = -2 \cdot \frac{1}{8} \).
• Simplify: this equals \( -\frac{2}{8} = -\frac{1}{4} \).

Evaluating derivatives is a useful step in understanding how a function behaves at certain points and can often be crucial for applications in physics, engineering, and other fields where calculus is applied.