Question
Find each deriuative 1) \( y=\sqrt{2 x-3} \)
Ask by Dawson Wheeler. in the United States
Feb 26,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The derivative of \( y = \sqrt{2x - 3} \) is \( y' = \frac{1}{\sqrt{2x - 3}} \).
Solution
To differentiate y = √(2x – 3), first rewrite it in exponent form:
y = (2x – 3)^(1/2)
Now apply the chain rule. Let u = 2x – 3, so that y = u^(1/2). Differentiate with respect to u:
dy/du = (1/2) u^(–1/2)
Next, differentiate u with respect to x:
du/dx = 2
Now, by the chain rule:
dy/dx = (dy/du) · (du/dx) = (1/2) u^(–1/2) · 2
Simplify:
dy/dx = u^(–1/2) = 1/√u
Finally, substitute back u = 2x – 3:
dy/dx = 1/√(2x – 3)
Thus, the derivative of y = √(2x – 3) is:
y' = 1/√(2x – 3)
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Simplify this solution Extra Insights
To find the derivative of \( y = \sqrt{2x - 3} \), we can use the chain rule. First, we rewrite it as \( y = (2x - 3)^{1/2} \). Then, applying the derivative: 1. Differentiate the outer function: \( \frac{1}{2} (2x - 3)^{-1/2} \). 2. Multiply by the derivative of the inner function: \( 2 \). So, the derivative is: \[ \frac{dy}{dx} = \frac{1}{2} (2x - 3)^{-1/2} \cdot 2 = \frac{1}{\sqrt{2x - 3}}. \] Thus, the final answer is: \[ \frac{dy}{dx} = \frac{1}{\sqrt{2x - 3}}. \]
