Question

The rate of change of $\sqrt{x^{2}+16}$ with respect to $\frac{x}{x-1}$ at $x=5$ is

• A. $\frac{-80}{\sqrt{41$
• B. $\frac{80}{\sqrt{41$
• C. $\frac{12}{5}$
• D. $\frac{-12}{5}$

Hint:

Logic Tip: When calculating derivative ratios at a specific point, it is usually faster to substitute the value of $x$ directly into $dy/dx$ and $dz/dx$ separately, then divide the resulting numerical values, rather than simplifying the full algebraic expression first!

Answer 1

The Correct Option is A

Concept:
To find the rate of change of one function $y=f(x)$ with respect to another function $z=g(x)$, we use the parametric differentiation rule: $$\frac{dy}{dz} = \frac{\frac{dy}{dx{\frac{dz}{dx$$
Step 1: Differentiate the first function (y) with respect to x.
Let $y = \sqrt{x^2+16}$. Using the chain rule: $$\frac{dy}{dx} = \frac{1}{2\sqrt{x^2+16 \cdot \frac{d}{dx}(x^2+16)$$ $$\frac{dy}{dx} = \frac{2x}{2\sqrt{x^2+16 = \frac{x}{\sqrt{x^2+16$$
Step 2: Differentiate the second function (z) with respect to x.
Let $z = \frac{x}{x-1}$. Using the quotient rule: $$\frac{dz}{dx} = \frac{(x-1)\cdot \frac{d}{dx}(x) - x \cdot \frac{d}{dx}(x-1)}{(x-1)^2}$$ $$\frac{dz}{dx} = \frac{(x-1)(1) - x(1)}{(x-1)^2} = \frac{x - 1 - x}{(x-1)^2} = \frac{-1}{(x-1)^2}$$
Step 3: Find the ratio dy/dz and evaluate at x = 5.
Now, calculate $\frac{dy}{dz}$: $$\frac{dy}{dz} = \frac{\frac{x}{\sqrt{x^2+16}{\frac{-1}{(x-1)^2 = \frac{-x(x-1)^2}{\sqrt{x^2+16$$ Substitute $x = 5$ into the expression: $$\left. \frac{dy}{dz} \right|_{x=5} = \frac{-5(5-1)^2}{\sqrt{5^2+16$$ $$= \frac{-5(4)^2}{\sqrt{25+16 = \frac{-5(16)}{\sqrt{41 = \frac{-80}{\sqrt{41$$