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Differentiate y = sqrt((1−2x)/(1+3x))

March 17, 2026Mathematics

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Answer

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

Graph of y = √((1 − 2x)/(1 + 3x))

Key points of y = √((1 − 2x)/(1 + 3x))
Point	x	y
y-intercept	0	1
x-intercept	0.5	0

6 steps

Rewrite $ y $ using exponential notation.
$y = (1 - 2x)^{\frac{1}{2}}$
Differentiate both sides with respect to $ x $ using the chain rule.
$\frac{dy}{dx} = \frac{1}{2}(1 - 2x)^{\frac{1}{2} - 1} \cdot \frac{d}{dx}(1 - 2x)$
Compute the derivative of the inner function.
$\frac{d}{dx}(1 - 2x) = -2$ $\frac{dy}{dx} = \frac{1}{2}(1 - 2x)^{-\frac{1}{2}} \cdot (-2)$
Simplify the constants.
$\frac{dy}{dx} = \frac{1}{2} \cdot (-2) \cdot (1 - 2x)^{-\frac{1}{2}}$ $\frac{dy}{dx} = - (1 - 2x)^{-\frac{1}{2}}$
Rewrite the negative exponent as a fraction.
$\frac{dy}{dx} = -\frac{1}{(1 - 2x)^{\frac{1}{2}}}$
Rewrite using radical notation.
$\frac{dy}{dx} = -\frac{1}{\sqrt{1 - 2x}}$ $\boxed{\dfrac{dy}{dx} = -\dfrac{1}{\sqrt{1 - 2x}}}$

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