Problem 24 - College Algebra and Calculus: An Applied Approach

Question

Use Example 6 as a model to find the derivative. $$ y=\frac{2}{3 x^{2}} $$

Step-by-Step Solution

Answer

The derivative of \(y = \frac{2}{3x^{2}}\) is \(-\frac{4}{3x^{3}}\) or \(-4/3x^{3}\).

Step 1: Rewrite the function

We rewrite the function \(y = \frac{2}{3x^{2}}\) as \(y = 2 \times \frac{1}{3} \times x^{-2}\), where we have simply broken down the constant multiplier (2/3) into two parts (2 and 1/3).

Step 2: Find the derivative

Next, by the power rule in calculus, the derivative of \(x^n\) is \(n*x^{n - 1}\). Applying this, the derivative of \(x^{-2}\) is -2*\(x^{-2-1} = -2*x^{-3}\). Multiplying this by our outside multiplier 2 * (1/3), we get that the derivative of our original equation \(y = 2x^{-2}\) is \(-2 * 2/3 * x^{-3}\), which simplifies to \(-4/3 * x^{-3}\).

Step 3: Simplify the derivative

The expression \(-4/3 * x^{-3}\) can be rewritten in positive exponent format as \(-4/3x^{3}\), and in fractional format as \(-\frac{4}{3x^{3}}\). Either of these can be the final answer, depending on the format your teacher prefers.