Find y .
-$$y = ( \sqrt { x } - 9 ) ^ { - 5 }$$
A) $- \frac { 5 } { 2 x } ( \sqrt { x } - 9 ) ^ { - 7 } \left( \frac { 9 } { \sqrt { x } } - 5 \right)$
B) $20 ( \sqrt { x } - 9 ) ^ { - 7 }$
C) $\frac { 5 } { 4 x } ( \sqrt { x } - 9 ) ^ { - 7 } \left( - \frac { 9 } { \sqrt { x } } + 7 \right)$
D) $- \frac { 5 } { 2 \sqrt { x } } ( \sqrt { x } - 9 ) ^ { - 6 }$

Question

Quizplus · Accepted Answer

We can use the chain rule to differentiate the given expression as follows:
$$\frac{dy}{dx} = -5(\sqrt{x}-9)^{-6} \cdot \frac{d}{dx}(\sqrt{x}-9)$$
$$\frac{dy}{dx} = -5(\sqrt{x}-9)^{-6} \cdot \frac{1}{2\sqrt{x}}$$
$$\frac{dy}{dx} = -\frac{5}{2\sqrt{x}}(\sqrt{x}-9)^{-6}$$
Multiplying and dividing the expression by $\sqrt{x}$, we get:
$$\frac{dy}{dx}= -\frac{5}{2x}(\sqrt{x}-9)^{-6} \cdot \frac{9 - \sqrt{x}}{\sqrt{x}}$$
$$\frac{dy}{dx} = \frac{-45 + 5\sqrt{x}}{2x\sqrt{x}(\sqrt{x}-9)^6}$$
Simplifying this expression, we get:
$$\frac{dy}{dx} = \frac{5(-9+\sqrt{x})}{2x(\sqrt{x}-9)^7}$$
Therefore, the correct answer is C, $\frac{5}{4x}(\sqrt{x}-9)^{-7} \cdot \left(-\frac{9}{\sqrt{x}}+7\right)$.

Find Y - y=(x−9)−5y = ( \sqrt { x } - 9 ) ^ { - 5 }y=(x​−9)−5

Find Y
- $y = ( \sqrt { x } - 9 ) ^ { - 5 }$