Problem 32 - College Algebra and Calculus: An Applied Approach

Question

Use the General Power Rule to find the derivative of the function. $$ g(x)=\sqrt{5-3 x} $$

Step-by-Step Solution

Answer

Using General Power Rule, the derivative of the function \(g(x)=\sqrt{5-3x}\) is \(g'(x) = -\frac{3}{2\sqrt{5-3x}}\).

Step 1: Rewrite using Power Rule

First, we write the function using a power instead of a root. Thus, \( g(x) = \sqrt{5-3x} \) is rewritten as \( g(x) = (5-3x)^{\frac{1}{2}} \).

Step 2: Apply Power Rule

Then, apply the Power Rule for differentiation, which states that the derivative of \( u^n \) is \( n \cdot u^{n-1} \cdot u' \). Here, \( u = 5-3x \) and \( n= \frac{1}{2} \). So, \( g'(x) = \frac{1}{2} \cdot (5-3x)^{\frac{1}{2} - 1} \cdot (-3) = - \frac{3}{2} \cdot (5-3x)^{-\frac{1}{2}} \).

Step 3: Simplify The Result

Finally, rewrite the negative exponents and simplify expression : \( g'(x) = -\frac{3}{2} \cdot \frac{1}{\sqrt{5-3x}} \), which simplifies to \( g'(x) = -\frac{3}{2\sqrt{5-3x}} \).