Question

question given the function y = (5 - x^2)/(6 - 5x), find dy/dx in simplified form. answer attempt 1 out of 2 dy/dx =

Explanation:

Step1: Identify quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $\frac{dy}{dx}=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^{2}}$. Here, $u = 5 - x^{2}$, $v=6 - 5x$.

Step2: Find $\frac{du}{dx}$ and $\frac{dv}{dx}$

Differentiate $u = 5 - x^{2}$ with respect to $x$: $\frac{du}{dx}=-2x$. Differentiate $v = 6 - 5x$ with respect to $x$: $\frac{dv}{dx}=-5$.

Step3: Apply the quotient - rule

Substitute $u$, $v$, $\frac{du}{dx}$, and $\frac{dv}{dx}$ into the quotient - rule formula.
\[

$$\begin{align*} \frac{dy}{dx}&=\frac{(6 - 5x)(-2x)-(5 - x^{2})(-5)}{(6 - 5x)^{2}}\\ &=\frac{-12x + 10x^{2}+25 - 5x^{2}}{(6 - 5x)^{2}}\\ &=\frac{5x^{2}-12x + 25}{(6 - 5x)^{2}} \end{align*}$$

\]

Answer:

$\frac{5x^{2}-12x + 25}{(6 - 5x)^{2}}$