Differentiate y = sqrt((1-2x)/(1+3x))

Mathematics
Differentiate y = sqrt((1-2x)/(1+3x))

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Answer

dydx=−11−2x\dfrac{dy}{dx} = -\dfrac{1}{\sqrt{1 - 2x}}

Step 1: Rewrite yy using exponential notation.

y=(1−2x)12y = (1 - 2x)^{\frac{1}{2}}

Step 2: Differentiate both sides with respect to xx using the chain rule.

dydx=12(1−2x)12−1⋅ddx(1−2x)\frac{dy}{dx} = \frac{1}{2}(1 - 2x)^{\frac{1}{2} - 1} \cdot \frac{d}{dx}(1 - 2x)

Step 3: Compute the derivative of the inner function.

ddx(1−2x)=−2\frac{d}{dx}(1 - 2x) = -2 dydx=12(1−2x)−12⋅(−2)\frac{dy}{dx} = \frac{1}{2}(1 - 2x)^{-\frac{1}{2}} \cdot (-2)

Step 4: Simplify the constants.

dydx=12⋅(−2)⋅(1−2x)−12\frac{dy}{dx} = \frac{1}{2} \cdot (-2) \cdot (1 - 2x)^{-\frac{1}{2}} dydx=−(1−2x)−12\frac{dy}{dx} = - (1 - 2x)^{-\frac{1}{2}}

Step 5: Rewrite the negative exponent as a fraction.

dydx=−1(1−2x)12\frac{dy}{dx} = -\frac{1}{(1 - 2x)^{\frac{1}{2}}}

Step 6: Rewrite using radical notation.

dydx=−11−2x\frac{dy}{dx} = -\frac{1}{\sqrt{1 - 2x}} \dfrac{dy{dx} = -\dfrac{1}{\sqrt{1 - 2x}}}
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Quick Answer

Rewrite y using exponential notation. y = (1 - 2x)^(1)/(2) Step 2: Differentiate both sides with respect to x using the chain rule.

Differentiate y = sqrt((1-2x)/(1+3x))
Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

ScanSolve AI Answer
Step 1: Rewrite y using exponential notation. y = (1 - 2x)^(1)/(2) Step 2: Differentiate both sides with respect to x using the chain rule. (dy)/(dx) = (1)/(2)(1 - 2x)^(1)/(2) - 1 · (d)/(dx)(1 - 2x) Step 3: Compute the derivative of the inner function. (d)/(dx)(1 - 2x) = -2 (dy)/(dx) = (1)/(2)(1 - 2x)^-(1)/(2) · (-2) Step 4: Simplify the constants. (dy)/(dx) = (1)/(2) · (-2) · (1 - 2x)^-(1)/(2) (dy)/(dx) = - (1 - 2x)^-(1)/(2) Step 5: Rewrite the negative exponent as a fraction. (dy)/(dx) = -(1)/((1 - 2x)^1)2 Step 6: Rewrite using radical notation. (dy)/(dx) = -(1)/(sqrt(1 - 2x)) (dy)/(dx) = -(1)/(sqrt(1 - 2x))